CINXE.COM

<!DOCTYPE html> <html lang="en" class="no-js"> <head> <meta charset="UTF-8"> <meta http-equiv="X-UA-Compatible" content="IE=edge"> <meta name="applicable-device" content="pc,mobile"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="robots" content="max-image-preview:large"> <meta name="access" content="Yes"> <meta name="360-site-verification" content="1268d79b5e96aecf3ff2a7dac04ad990" /> <title>Frames in cofibration categories | Journal of Homotopy and Related Structures </title> <meta name="twitter:site" content="@SpringerLink"/> <meta name="twitter:card" content="summary_large_image"/> <meta name="twitter:image:alt" content="Content cover image"/> <meta name="twitter:title" content="Frames in cofibration categories"/> <meta name="twitter:description" content="Journal of Homotopy and Related Structures - We introduce the quasicategory of frames of a cofibration category, i.e. a new model of the $$(\infty ,1)$$ -category associated with a cofibration..."/> <meta name="twitter:image" content="https://static-content.springer.com/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figa_HTML.gif"/> <meta name="journal_id" content="40062"/> <meta name="dc.title" content="Frames in cofibration categories"/> <meta name="dc.source" content="Journal of Homotopy and Related Structures 2016 12:3"/> <meta name="dc.format" content="text/html"/> <meta name="dc.publisher" content="Springer"/> <meta name="dc.date" content="2016-07-13"/> <meta name="dc.type" content="OriginalPaper"/> <meta name="dc.language" content="En"/> <meta name="dc.copyright" content="2016 Tbilisi Centre for Mathematical Sciences"/> <meta name="dc.rights" content="2016 Tbilisi Centre for Mathematical Sciences"/> <meta name="dc.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="dc.description" content="We introduce the quasicategory of frames of a cofibration category, i.e. a new model of the $$(\infty ,1)$$ -category associated with a cofibration category."/> <meta name="prism.issn" content="1512-2891"/> <meta name="prism.publicationName" content="Journal of Homotopy and Related Structures"/> <meta name="prism.publicationDate" content="2016-07-13"/> <meta name="prism.volume" content="12"/> <meta name="prism.number" content="3"/> <meta name="prism.section" content="OriginalPaper"/> <meta name="prism.startingPage" content="577"/> <meta name="prism.endingPage" content="616"/> <meta name="prism.copyright" content="2016 Tbilisi Centre for Mathematical Sciences"/> <meta name="prism.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="prism.url" content="https://link.springer.com/article/10.1007/s40062-016-0139-x"/> <meta name="prism.doi" content="doi:10.1007/s40062-016-0139-x"/> <meta name="citation_pdf_url" content="https://link.springer.com/content/pdf/10.1007/s40062-016-0139-x.pdf"/> <meta name="citation_fulltext_html_url" content="https://link.springer.com/article/10.1007/s40062-016-0139-x"/> <meta name="citation_journal_title" content="Journal of Homotopy and Related Structures"/> <meta name="citation_journal_abbrev" content="J. Homotopy Relat. Struct."/> <meta name="citation_publisher" content="Springer Berlin Heidelberg"/> <meta name="citation_issn" content="1512-2891"/> <meta name="citation_title" content="Frames in cofibration categories"/> <meta name="citation_volume" content="12"/> <meta name="citation_issue" content="3"/> <meta name="citation_publication_date" content="2017/09"/> <meta name="citation_online_date" content="2016/07/13"/> <meta name="citation_firstpage" content="577"/> <meta name="citation_lastpage" content="616"/> <meta name="citation_article_type" content="Article"/> <meta name="citation_fulltext_world_readable" content=""/> <meta name="citation_language" content="en"/> <meta name="dc.identifier" content="doi:10.1007/s40062-016-0139-x"/> <meta name="DOI" content="10.1007/s40062-016-0139-x"/> <meta name="size" content="1304217"/> <meta name="citation_doi" content="10.1007/s40062-016-0139-x"/> <meta name="citation_springer_api_url" content="http://api.springer.com/xmldata/jats?q=doi:10.1007/s40062-016-0139-x&api_key="/> <meta name="description" content="We introduce the quasicategory of frames of a cofibration category, i.e. a new model of the $$(\infty ,1)$$ -category associated with a cofibration categor"/> <meta name="dc.creator" content="Szumiło, Karol"/> <meta name="dc.subject" content="Algebraic Topology"/> <meta name="dc.subject" content="Algebra"/> <meta name="dc.subject" content="Functional Analysis"/> <meta name="dc.subject" content="Number Theory"/> <meta name="citation_reference" content="citation_journal_title=Math. Struct. Comput. Sci.; citation_title=Homotopy limits in type theory; citation_author=J Avigad, K Kapulkin, PL Lumsdaine; citation_volume=25; citation_issue=5; citation_publication_date=2015; citation_pages=1040-1070; citation_doi=10.1017/S0960129514000498; citation_id=CR1"/> <meta name="citation_reference" content="Borceux, F.: Handbook of categorical algebra. 1. Encyclopedia of Mathematics and its Applications, vol. 50, Cambridge University Press, Cambridge. Basic category theory (1994)"/> <meta name="citation_reference" content="citation_journal_title=Trans. Am. Math. Soc.; citation_title=Abstract homotopy theory and generalized sheaf cohomology; citation_author=KS Brown; citation_volume=186; citation_publication_date=1973; citation_pages=419-458; citation_doi=10.1090/S0002-9947-1973-0341469-9; citation_id=CR3"/> <meta name="citation_reference" content="Cordier, J.M.: Sur la notion de diagramme homotopiquement cohérent. Cahiers Topologie Géom. Différentielle 23(1), 93–112 (1982) (French). Third Colloquium on Categories, Part VI (Amiens, 1980)"/> <meta name="citation_reference" content="Dwyer, W.G., Kan, D.M.: Simplicial localizations of categories. J. Pure Appl. Algebra 17(3), 267–284 (1980)"/> <meta name="citation_reference" content="citation_journal_title=J. Pure Appl. Algebra; citation_title=Calculating simplicial localizations; citation_author=WG Dwyer, DM Kan; citation_volume=18; citation_issue=1; citation_publication_date=1980; citation_pages=17-35; citation_doi=10.1016/0022-4049(80)90113-9; citation_id=CR6"/> <meta name="citation_reference" content="citation_title=Calculus of Fractions and Homotopy Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete; citation_publication_date=1967; citation_id=CR7; citation_author=P Gabriel; citation_author=M Zisman; citation_publisher=Springer-Verlag New York Inc"/> <meta name="citation_reference" content="citation_title=Model categories, Mathematical Surveys and Monographs; citation_publication_date=1999; citation_id=CR8; citation_author=M Hovey; citation_publisher=American Mathematical Society"/> <meta name="citation_reference" content="Joyal, A.: Quasi-categories and Kan complexes. J. Pure Appl. Algebra 175(1–3), 207–222 (2002). doi: 10.1016/S0022-4049(02)00135-4 . Special volume celebrating the 70th birthday of Professor Max Kelly. MR1935979"/> <meta name="citation_reference" content="Joyal, A.: The Theory of Quasi-Categories and its Applications. Quadern 45, Vol. II, Centre de Recerca Matemàtica Barcelona (2008)"/> <meta name="citation_reference" content="Joyal, A., Tierney, M.: Quasi-categories vs Segal spaces. Categories in algebra, geometry and mathematical physics, Contemp. Math., vol. 431, pp. 277–326. Amer. Math. Soc., Providence, RI (2007)"/> <meta name="citation_reference" content="Kapulkin, K.: Locally Cartesian Closed Quasicategories From Type Theory. arXiv:1507.02648 "/> <meta name="citation_reference" content="Kapulkin, K., Szumiło, K.: Quasicategories of frames of cofibration categories. Appl. Categ. Struct., 1–25 (2016)"/> <meta name="citation_reference" content="Răadulescu-Banu, A.: Cofibrations in Homotopy Theory (2006). http://arxiv.org/abs/math/0610009v4 "/> <meta name="citation_reference" content="Rezk, C.: A model for the homotopy theory of homotopy theory. Trans. Amer. Math. Soc. 353(3), 973–1007 (2001) (electronic)"/> <meta name="citation_reference" content="Rourke, C.P., Sanderson, B.J.: $$\triangle $$ -sets. I. Homotopy theory. Quart. J. Math. Oxford Ser. 22(2), 321–338 (1971)"/> <meta name="citation_reference" content="citation_journal_title=Theory Appl. Categ.; citation_title=The theory and practice of Reedy categories; citation_author=E Riehl, D Verity; citation_volume=29; citation_publication_date=2014; citation_pages=256-301; citation_id=CR17"/> <meta name="citation_reference" content="citation_journal_title=J. Topol.; citation_title=The p-order of topological triangulated categories; citation_author=S Schwede; citation_volume=6; citation_issue=4; citation_publication_date=2013; citation_pages=868-914; citation_doi=10.1112/jtopol/jtt018; citation_id=CR18"/> <meta name="citation_reference" content="Szumiło, K.: Two Models for the Homotopy Theory of Cocomplete Homotopy Theories. Ph.D. Thesis, Rheinische Friedrich-Wilhelms-Universität Bonn (2014). http://hss.ulb.uni-bonn.de/2014/3692/3692.htm "/> <meta name="citation_reference" content="Szumiło, K.: Two Models for the Homotopy Theory of Cocomplete Homotopy Theories (2014). arXiv:1411.0303 "/> <meta name="citation_reference" content="Szumiło, K.: Homotopy theory of cofibration categories. Homol. Homotop. Appl (to appear)"/> <meta name="citation_reference" content="Szumiło, K.: Homotopy theory of cocomplete quasicategories. Algebraic Geom Topol (to appear)"/> <meta name="citation_author" content="Szumiło, Karol"/> <meta name="citation_author_email" content="kszumilo@uwo.ca"/> <meta name="citation_author_institution" content="Department of Mathematics, University of Western Ontario, London, Canada"/> <meta name="format-detection" content="telephone=no"/> <meta name="citation_cover_date" content="2017/09/01"/> <meta property="og:url" content="https://link.springer.com/article/10.1007/s40062-016-0139-x"/> <meta property="og:type" content="article"/> <meta property="og:site_name" content="SpringerLink"/> <meta property="og:title" content="Frames in cofibration categories - Journal of Homotopy and Related Structures"/> <meta property="og:description" content="We introduce the quasicategory of frames of a cofibration category, i.e. a new model of the $$(\infty ,1)$$ ( ∞ , 1 ) -category associated with a cofibration category."/> <meta property="og:image" content="https://static-content.springer.com/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figa_HTML.gif"/> <meta name="format-detection" content="telephone=no"> <link rel="apple-touch-icon" sizes="180x180" href=/oscar-static/img/favicons/darwin/apple-touch-icon-92e819bf8a.png> <link rel="icon" type="image/png" sizes="192x192" href=/oscar-static/img/favicons/darwin/android-chrome-192x192-6f081ca7e5.png> <link rel="icon" type="image/png" sizes="32x32" href=/oscar-static/img/favicons/darwin/favicon-32x32-1435da3e82.png> <link rel="icon" type="image/png" sizes="16x16" href=/oscar-static/img/favicons/darwin/favicon-16x16-ed57f42bd2.png> <link rel="shortcut icon" data-test="shortcut-icon" href=/oscar-static/img/favicons/darwin/favicon-c6d59aafac.ico> <meta name="theme-color" content="#e6e6e6">   <link rel="stylesheet" media="print" href=/oscar-static/app-springerlink/css/print-b8af42253b.css> <style> html{text-size-adjust:100%;line-height:1.15}body{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;line-height:1.8;margin:0}details,main{display:block}h1{font-size:2em;margin:.67em 0}a{background-color:transparent;color:#025e8d}sub{bottom:-.25em;font-size:75%;line-height:0;position:relative;vertical-align:baseline}img{border:0;height:auto;max-width:100%;vertical-align:middle}button,input{font-family:inherit;font-size:100%;line-height:1.15;margin:0;overflow:visible}button{text-transform:none}[type=button],[type=submit],button{-webkit-appearance:button}[type=search]{-webkit-appearance:textfield;outline-offset:-2px}summary{display:list-item}[hidden]{display:none}button{cursor:pointer}svg{height:1rem;width:1rem} </style> <style>@media only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark) { body{background:#fff;color:#222;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;line-height:1.8;min-height:100%}a{color:#025e8d;text-decoration:underline;text-decoration-skip-ink:auto}button{cursor:pointer}img{border:0;height:auto;max-width:100%;vertical-align:middle}html{box-sizing:border-box;font-size:100%;height:100%;overflow-y:scroll}h1{font-size:2.25rem}h2{font-size:1.75rem}h1,h2,h4{font-weight:700;line-height:1.2}h4{font-size:1.25rem}body{font-size:1.125rem}*{box-sizing:inherit}p{margin-bottom:2rem;margin-top:0}p:last-of-type{margin-bottom:0}.c-ad{text-align:center}@media only screen and (min-width:480px){.c-ad{padding:8px}}.c-ad--728x90{display:none}.c-ad--728x90 .c-ad__inner{min-height:calc(1.5em + 94px)}@media only screen and (min-width:876px){.js .c-ad--728x90{display:none}}.c-ad__label{color:#333;font-size:.875rem;font-weight:400;line-height:1.5;margin-bottom:4px}.c-ad__label,.c-status-message{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.c-status-message{align-items:center;box-sizing:border-box;display:flex;position:relative;width:100%}.c-status-message :last-child{margin-bottom:0}.c-status-message--boxed{background-color:#fff;border:1px solid #ccc;line-height:1.4;padding:16px}.c-status-message__heading{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:.875rem;font-weight:700}.c-status-message__icon{fill:currentcolor;display:inline-block;flex:0 0 auto;height:1.5em;margin-right:8px;transform:translate(0);vertical-align:text-top;width:1.5em}.c-status-message__icon--top{align-self:flex-start}.c-status-message--info .c-status-message__icon{color:#003f8d}.c-status-message--boxed.c-status-message--info{border-bottom:4px solid #003f8d}.c-status-message--error .c-status-message__icon{color:#c40606}.c-status-message--boxed.c-status-message--error{border-bottom:4px solid #c40606}.c-status-message--success .c-status-message__icon{color:#00b8b0}.c-status-message--boxed.c-status-message--success{border-bottom:4px solid #00b8b0}.c-status-message--warning .c-status-message__icon{color:#edbc53}.c-status-message--boxed.c-status-message--warning{border-bottom:4px solid #edbc53}.eds-c-header{background-color:#fff;border-bottom:2px solid #01324b;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:1rem;line-height:1.5;padding:8px 0 0}.eds-c-header__container{align-items:center;display:flex;flex-wrap:nowrap;gap:8px 16px;justify-content:space-between;margin:0 auto 8px;max-width:1280px;padding:0 8px;position:relative}.eds-c-header__nav{border-top:2px solid #c5e0f4;padding-top:4px;position:relative}.eds-c-header__nav-container{align-items:center;display:flex;flex-wrap:wrap;margin:0 auto 4px;max-width:1280px;padding:0 8px;position:relative}.eds-c-header__nav-container>:not(:last-child){margin-right:32px}.eds-c-header__link-container{align-items:center;display:flex;flex:1 0 auto;gap:8px 16px;justify-content:space-between}.eds-c-header__list{list-style:none;margin:0;padding:0}.eds-c-header__list-item{font-weight:700;margin:0 auto;max-width:1280px;padding:8px}.eds-c-header__list-item:not(:last-child){border-bottom:2px solid #c5e0f4}.eds-c-header__item{color:inherit}@media only screen and (min-width:768px){.eds-c-header__item--menu{display:none;visibility:hidden}.eds-c-header__item--menu:first-child+*{margin-block-start:0}}.eds-c-header__item--inline-links{display:none;visibility:hidden}@media only screen and (min-width:768px){.eds-c-header__item--inline-links{display:flex;gap:16px 16px;visibility:visible}}.eds-c-header__item--divider:before{border-left:2px solid #c5e0f4;content:"";height:calc(100% - 16px);margin-left:-15px;position:absolute;top:8px}.eds-c-header__brand{padding:16px 8px}.eds-c-header__brand a{display:block;line-height:1;text-decoration:none}.eds-c-header__brand img{height:1.5rem;width:auto}.eds-c-header__link{color:inherit;display:inline-block;font-weight:700;padding:16px 8px;position:relative;text-decoration-color:transparent;white-space:nowrap;word-break:normal}.eds-c-header__icon{fill:currentcolor;display:inline-block;font-size:1.5rem;height:1em;transform:translate(0);vertical-align:bottom;width:1em}.eds-c-header__icon+*{margin-left:8px}.eds-c-header__expander{background-color:#f0f7fc}.eds-c-header__search{display:block;padding:24px 0}@media only screen and (min-width:768px){.eds-c-header__search{max-width:70%}}.eds-c-header__search-container{position:relative}.eds-c-header__search-label{color:inherit;display:inline-block;font-weight:700;margin-bottom:8px}.eds-c-header__search-input{background-color:#fff;border:1px solid #000;padding:8px 48px 8px 8px;width:100%}.eds-c-header__search-button{background-color:transparent;border:0;color:inherit;height:100%;padding:0 8px;position:absolute;right:0}.has-tethered.eds-c-header__expander{border-bottom:2px solid #01324b;left:0;margin-top:-2px;top:100%;width:100%;z-index:10}@media only screen and (min-width:768px){.has-tethered.eds-c-header__expander--menu{display:none;visibility:hidden}}.has-tethered .eds-c-header__heading{display:none;visibility:hidden}.has-tethered .eds-c-header__heading:first-child+*{margin-block-start:0}.has-tethered .eds-c-header__search{margin:auto}.eds-c-header__heading{margin:0 auto;max-width:1280px;padding:16px 16px 0}.eds-c-pagination{align-items:center;display:flex;flex-wrap:wrap;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:.875rem;gap:16px 0;justify-content:center;line-height:1.4;list-style:none;margin:0;padding:32px 0}@media only screen and (min-width:480px){.eds-c-pagination{padding:32px 16px}}.eds-c-pagination__item{margin-right:8px}.eds-c-pagination__item--prev{margin-right:16px}.eds-c-pagination__item--next .eds-c-pagination__link,.eds-c-pagination__item--prev .eds-c-pagination__link{padding:16px 8px}.eds-c-pagination__item--next{margin-left:8px}.eds-c-pagination__item:last-child{margin-right:0}.eds-c-pagination__link{align-items:center;color:#222;cursor:pointer;display:inline-block;font-size:1rem;margin:0;padding:16px 24px;position:relative;text-align:center;transition:all .2s ease 0s}.eds-c-pagination__link:visited{color:#222}.eds-c-pagination__link--disabled{border-color:#555;color:#555;cursor:default}.eds-c-pagination__link--active{background-color:#01324b;background-image:none;border-radius:8px;color:#fff}.eds-c-pagination__link--active:focus,.eds-c-pagination__link--active:hover,.eds-c-pagination__link--active:visited{color:#fff}.eds-c-pagination__link-container{align-items:center;display:flex}.eds-c-pagination__icon{fill:#222;height:1.5rem;width:1.5rem}.eds-c-pagination__icon--disabled{fill:#555}.eds-c-pagination__visually-hidden{clip:rect(0,0,0,0);border:0;clip-path:inset(50%);height:1px;overflow:hidden;padding:0;position:absolute!important;white-space:nowrap;width:1px}.c-breadcrumbs{color:#333;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:1rem;list-style:none;margin:0;padding:0}.c-breadcrumbs>li{display:inline}svg.c-breadcrumbs__chevron{fill:#333;height:10px;margin:0 .25rem;width:10px}.c-breadcrumbs--contrast,.c-breadcrumbs--contrast .c-breadcrumbs__link{color:#fff}.c-breadcrumbs--contrast svg.c-breadcrumbs__chevron{fill:#fff}@media only screen and (max-width:479px){.c-breadcrumbs .c-breadcrumbs__item{display:none}.c-breadcrumbs .c-breadcrumbs__item:last-child,.c-breadcrumbs .c-breadcrumbs__item:nth-last-child(2){display:inline}}.c-skip-link{background:#01324b;bottom:auto;color:#fff;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:1rem;padding:8px;position:absolute;text-align:center;transform:translateY(-100%);width:100%;z-index:9999}@media (prefers-reduced-motion:reduce){.c-skip-link{transition:top .3s ease-in-out 0s}}@media print{.c-skip-link{display:none}}.c-skip-link:active,.c-skip-link:hover,.c-skip-link:link,.c-skip-link:visited{color:#fff}.c-skip-link:focus{transform:translateY(0)}.l-with-sidebar{display:flex;flex-wrap:wrap}.l-with-sidebar>*{margin:0}.l-with-sidebar__sidebar{flex-basis:var(--with-sidebar--basis,400px);flex-grow:1}.l-with-sidebar>:not(.l-with-sidebar__sidebar){flex-basis:0px;flex-grow:999;min-width:var(--with-sidebar--min,53%)}.l-with-sidebar>:first-child{padding-right:4rem}@supports (gap:1em){.l-with-sidebar>:first-child{padding-right:0}.l-with-sidebar{gap:var(--with-sidebar--gap,4rem)}}.c-header__link{color:inherit;display:inline-block;font-weight:700;padding:16px 8px;position:relative;text-decoration-color:transparent;white-space:nowrap;word-break:normal}.app-masthead__colour-4{--background-color:#ff9500;--gradient-light:rgba(0,0,0,.5);--gradient-dark:rgba(0,0,0,.8)}.app-masthead{background:var(--background-color,#0070a8);position:relative}.app-masthead:after{background:radial-gradient(circle at top right,var(--gradient-light,rgba(0,0,0,.4)),var(--gradient-dark,rgba(0,0,0,.7)));bottom:0;content:"";left:0;position:absolute;right:0;top:0}@media only screen and (max-width:479px){.app-masthead:after{background:linear-gradient(225deg,var(--gradient-light,rgba(0,0,0,.4)),var(--gradient-dark,rgba(0,0,0,.7)))}}.app-masthead__container{color:var(--masthead-color,#fff);margin:0 auto;max-width:1280px;padding:0 16px;position:relative;z-index:1}.u-button{align-items:center;background-color:#01324b;background-image:none;border:4px solid transparent;border-radius:32px;cursor:pointer;display:inline-flex;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:.875rem;font-weight:700;justify-content:center;line-height:1.3;margin:0;padding:16px 32px;position:relative;transition:all .2s ease 0s;width:auto}.u-button svg,.u-button--contrast svg,.u-button--primary svg,.u-button--secondary svg,.u-button--tertiary svg{fill:currentcolor}.u-button,.u-button:visited{color:#fff}.u-button,.u-button:hover{box-shadow:0 0 0 1px #01324b;text-decoration:none}.u-button:hover{border:4px solid #fff}.u-button:focus{border:4px solid #fc0;box-shadow:none;outline:0;text-decoration:none}.u-button:focus,.u-button:hover{background-color:#fff;background-image:none;color:#01324b}.app-masthead--pastel .c-pdf-download .u-button--primary:focus svg path,.app-masthead--pastel .c-pdf-download .u-button--primary:hover svg path,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:focus svg path,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:hover svg path,.u-button--primary:focus svg path,.u-button--primary:hover svg path,.u-button:focus svg path,.u-button:hover svg path{fill:#01324b}.u-button--primary{background-color:#01324b;background-image:none;border:4px solid transparent;box-shadow:0 0 0 1px #01324b;color:#fff;font-weight:700}.u-button--primary:visited{color:#fff}.u-button--primary:hover{border:4px solid #fff;box-shadow:0 0 0 1px #01324b;text-decoration:none}.u-button--primary:focus{border:4px solid #fc0;box-shadow:none;outline:0;text-decoration:none}.u-button--primary:focus,.u-button--primary:hover{background-color:#fff;background-image:none;color:#01324b}.u-button--secondary{background-color:#fff;border:4px solid #fff;color:#01324b;font-weight:700}.u-button--secondary:visited{color:#01324b}.u-button--secondary:hover{border:4px solid #01324b;box-shadow:none}.u-button--secondary:focus,.u-button--secondary:hover{background-color:#01324b;color:#fff}.app-masthead--pastel .c-pdf-download .u-button--secondary:focus svg path,.app-masthead--pastel .c-pdf-download .u-button--secondary:hover svg path,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary:focus svg path,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary:hover svg path,.u-button--secondary:focus svg path,.u-button--secondary:hover svg path,.u-button--tertiary:focus svg path,.u-button--tertiary:hover svg path{fill:#fff}.u-button--tertiary{background-color:#ebf1f5;border:4px solid transparent;box-shadow:none;color:#666;font-weight:700}.u-button--tertiary:visited{color:#666}.u-button--tertiary:hover{border:4px solid #01324b;box-shadow:none}.u-button--tertiary:focus,.u-button--tertiary:hover{background-color:#01324b;color:#fff}.u-button--contrast{background-color:transparent;background-image:none;color:#fff;font-weight:400}.u-button--contrast:visited{color:#fff}.u-button--contrast,.u-button--contrast:focus,.u-button--contrast:hover{border:4px solid #fff}.u-button--contrast:focus,.u-button--contrast:hover{background-color:#fff;background-image:none;color:#000}.u-button--contrast:focus svg path,.u-button--contrast:hover svg path{fill:#000}.u-button--disabled,.u-button:disabled{background-color:transparent;background-image:none;border:4px solid #ccc;color:#000;cursor:default;font-weight:400;opacity:.7}.u-button--disabled svg,.u-button:disabled svg{fill:currentcolor}.u-button--disabled:visited,.u-button:disabled:visited{color:#000}.u-button--disabled:focus,.u-button--disabled:hover,.u-button:disabled:focus,.u-button:disabled:hover{border:4px solid #ccc;text-decoration:none}.u-button--disabled:focus,.u-button--disabled:hover,.u-button:disabled:focus,.u-button:disabled:hover{background-color:transparent;background-image:none;color:#000}.u-button--disabled:focus svg path,.u-button--disabled:hover svg path,.u-button:disabled:focus svg path,.u-button:disabled:hover svg path{fill:#000}.u-button--small,.u-button--xsmall{font-size:.875rem;padding:2px 8px}.u-button--small{padding:8px 16px}.u-button--large{font-size:1.125rem;padding:10px 35px}.u-button--full-width{display:flex;width:100%}.u-button--icon-left svg{margin-right:8px}.u-button--icon-right svg{margin-left:8px}.u-clear-both{clear:both}.u-container{margin:0 auto;max-width:1280px;padding:0 16px}.u-justify-content-space-between{justify-content:space-between}.u-display-none{display:none}.js .u-js-hide,.u-hide{display:none;visibility:hidden}.u-visually-hidden{clip:rect(0,0,0,0);border:0;clip-path:inset(50%);height:1px;overflow:hidden;padding:0;position:absolute!important;white-space:nowrap;width:1px}.u-icon{fill:currentcolor;display:inline-block;height:1em;transform:translate(0);vertical-align:text-top;width:1em}.u-list-reset{list-style:none;margin:0;padding:0}.u-ma-16{margin:16px}.u-mt-0{margin-top:0}.u-mt-24{margin-top:24px}.u-mt-32{margin-top:32px}.u-mb-8{margin-bottom:8px}.u-mb-32{margin-bottom:32px}.u-button-reset{background-color:transparent;border:0;padding:0}.u-sans-serif{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.u-serif{font-family:Merriweather,serif}h1,h2,h4{-webkit-font-smoothing:antialiased}p{overflow-wrap:break-word;word-break:break-word}.u-h4{font-size:1.25rem;font-weight:700;line-height:1.2}.u-mbs-0{margin-block-start:0!important}.c-article-header{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.c-article-identifiers{color:#6f6f6f;display:flex;flex-wrap:wrap;font-size:1rem;line-height:1.3;list-style:none;margin:0 0 8px;padding:0}.c-article-identifiers__item{border-right:1px solid #6f6f6f;list-style:none;margin-right:8px;padding-right:8px}.c-article-identifiers__item:last-child{border-right:0;margin-right:0;padding-right:0}@media only screen and (min-width:876px){.c-article-title{font-size:1.875rem;line-height:1.2}}.c-article-author-list{display:inline;font-size:1rem;list-style:none;margin:0 8px 0 0;padding:0;width:100%}.c-article-author-list__item{display:inline;padding-right:0}.c-article-author-list__show-more{display:none;margin-right:4px}.c-article-author-list__button,.js .c-article-author-list__item--hide,.js .c-article-author-list__show-more{display:none}.js .c-article-author-list--long .c-article-author-list__show-more,.js .c-article-author-list--long+.c-article-author-list__button{display:inline}@media only screen and (max-width:767px){.js .c-article-author-list__item--hide-small-screen{display:none}.js .c-article-author-list--short .c-article-author-list__show-more,.js .c-article-author-list--short+.c-article-author-list__button{display:inline}}#uptodate-client,.js .c-article-author-list--expanded .c-article-author-list__show-more{display:none!important}.js .c-article-author-list--expanded .c-article-author-list__item--hide-small-screen{display:inline!important}.c-article-author-list__button,.c-button-author-list{background:#ebf1f5;border:4px solid #ebf1f5;border-radius:20px;color:#666;font-size:.875rem;line-height:1.4;padding:2px 11px 2px 8px;text-decoration:none}.c-article-author-list__button svg,.c-button-author-list svg{margin:1px 4px 0 0}.c-article-author-list__button:hover,.c-button-author-list:hover{background:#025e8d;border-color:transparent;color:#fff}.c-article-body .c-article-access-provider{padding:8px 16px}.c-article-body .c-article-access-provider,.c-notes{border:1px solid #d5d5d5;border-image:initial;border-left:none;border-right:none;margin:24px 0}.c-article-body .c-article-access-provider__text{color:#555}.c-article-body .c-article-access-provider__text,.c-notes__text{font-size:1rem;margin-bottom:0;padding-bottom:2px;padding-top:2px;text-align:center}.c-article-body .c-article-author-affiliation__address{color:inherit;font-weight:700;margin:0}.c-article-body .c-article-author-affiliation__authors-list{list-style:none;margin:0;padding:0}.c-article-body .c-article-author-affiliation__authors-item{display:inline;margin-left:0}.c-article-authors-search{margin-bottom:24px;margin-top:0}.c-article-authors-search__item,.c-article-authors-search__title{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.c-article-authors-search__title{color:#626262;font-size:1.05rem;font-weight:700;margin:0;padding:0}.c-article-authors-search__item{font-size:1rem}.c-article-authors-search__text{margin:0}.c-code-block{border:1px solid #fff;font-family:monospace;margin:0 0 24px;padding:20px}.c-code-block__heading{font-weight:400;margin-bottom:16px}.c-code-block__line{display:block;overflow-wrap:break-word;white-space:pre-wrap}.c-article-share-box{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;margin-bottom:24px}.c-article-share-box__description{font-size:1rem;margin-bottom:8px}.c-article-share-box__no-sharelink-info{font-size:.813rem;font-weight:700;margin-bottom:24px;padding-top:4px}.c-article-share-box__only-read-input{border:1px solid #d5d5d5;box-sizing:content-box;display:inline-block;font-size:.875rem;font-weight:700;height:24px;margin-bottom:8px;padding:8px 10px}.c-article-share-box__additional-info{color:#626262;font-size:.813rem}.c-article-share-box__button{background:#fff;box-sizing:content-box;text-align:center}.c-article-share-box__button--link-like{background-color:transparent;border:0;color:#025e8d;cursor:pointer;font-size:.875rem;margin-bottom:8px;margin-left:10px}.c-article-associated-content__container .c-article-associated-content__collection-label{font-size:.875rem;line-height:1.4}.c-article-associated-content__container .c-article-associated-content__collection-title{line-height:1.3}.c-reading-companion{clear:both;min-height:389px}.c-reading-companion__figures-list,.c-reading-companion__references-list{list-style:none;min-height:389px;padding:0}.c-reading-companion__references-list--numeric{list-style:decimal inside}.c-reading-companion__figure-item{border-top:1px solid #d5d5d5;font-size:1rem;padding:16px 8px 16px 0}.c-reading-companion__figure-item:first-child{border-top:none;padding-top:8px}.c-reading-companion__reference-item{font-size:1rem}.c-reading-companion__reference-item:first-child{border-top:none}.c-reading-companion__reference-item a{word-break:break-word}.c-reading-companion__reference-citation{display:inline}.c-reading-companion__reference-links{font-size:.813rem;font-weight:700;list-style:none;margin:8px 0 0;padding:0;text-align:right}.c-reading-companion__reference-links>a{display:inline-block;padding-left:8px}.c-reading-companion__reference-links>a:first-child{display:inline-block;padding-left:0}.c-reading-companion__figure-title{display:block;font-size:1.25rem;font-weight:700;line-height:1.2;margin:0 0 8px}.c-reading-companion__figure-links{display:flex;justify-content:space-between;margin:8px 0 0}.c-reading-companion__figure-links>a{align-items:center;display:flex}.c-article-section__figure-caption{display:block;margin-bottom:8px;word-break:break-word}.c-article-section__figure .video,p.app-article-masthead__access--above-download{margin:0 0 16px}.c-article-section__figure-description{font-size:1rem}.c-article-section__figure-description>*{margin-bottom:0}.c-cod{display:block;font-size:1rem;width:100%}.c-cod__form{background:#ebf0f3}.c-cod__prompt{font-size:1.125rem;line-height:1.3;margin:0 0 24px}.c-cod__label{display:block;margin:0 0 4px}.c-cod__row{display:flex;margin:0 0 16px}.c-cod__row:last-child{margin:0}.c-cod__input{border:1px solid #d5d5d5;border-radius:2px;flex-shrink:0;margin:0;padding:13px}.c-cod__input--submit{background-color:#025e8d;border:1px solid #025e8d;color:#fff;flex-shrink:1;margin-left:8px;transition:background-color .2s ease-out 0s,color .2s ease-out 0s}.c-cod__input--submit-single{flex-basis:100%;flex-shrink:0;margin:0}.c-cod__input--submit:focus,.c-cod__input--submit:hover{background-color:#fff;color:#025e8d}.save-data .c-article-author-institutional-author__sub-division,.save-data .c-article-equation__number,.save-data .c-article-figure-description,.save-data .c-article-fullwidth-content,.save-data .c-article-main-column,.save-data .c-article-satellite-article-link,.save-data .c-article-satellite-subtitle,.save-data .c-article-table-container,.save-data .c-blockquote__body,.save-data .c-code-block__heading,.save-data .c-reading-companion__figure-title,.save-data .c-reading-companion__reference-citation,.save-data .c-site-messages--nature-briefing-email-variant .serif,.save-data .c-site-messages--nature-briefing-email-variant.serif,.save-data .serif,.save-data .u-serif,.save-data h1,.save-data h2,.save-data h3{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.c-pdf-download__link{display:flex;flex:1 1 0%;padding:13px 24px}.c-pdf-download__link:hover{text-decoration:none}@media only screen and (min-width:768px){.c-context-bar--sticky .c-pdf-download__link{align-items:center;flex:1 1 183px}}@media only screen and (max-width:320px){.c-context-bar--sticky .c-pdf-download__link{padding:16px}}.c-article-body .c-article-recommendations-list,.c-book-body .c-article-recommendations-list{display:flex;flex-direction:row;gap:16px 16px;margin:0;max-width:100%;padding:16px 0 0}.c-article-body .c-article-recommendations-list__item,.c-book-body .c-article-recommendations-list__item{flex:1 1 0%}@media only screen and (max-width:767px){.c-article-body .c-article-recommendations-list,.c-book-body .c-article-recommendations-list{flex-direction:column}}.c-article-body .c-article-recommendations-card__authors{display:none;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:.875rem;line-height:1.5;margin:0 0 8px}@media only screen and (max-width:767px){.c-article-body .c-article-recommendations-card__authors{display:block;margin:0}}.c-article-body .c-article-history{margin-top:24px}.app-article-metrics-bar p{margin:0}.app-article-masthead{display:flex;flex-direction:column;gap:16px 16px;padding:16px 0 24px}.app-article-masthead__info{display:flex;flex-direction:column;flex-grow:1}.app-article-masthead__brand{border-top:1px solid hsla(0,0%,100%,.8);display:flex;flex-direction:column;flex-shrink:0;gap:8px 8px;min-height:96px;padding:16px 0 0}.app-article-masthead__brand img{border:1px solid #fff;border-radius:8px;box-shadow:0 4px 15px 0 hsla(0,0%,50%,.25);height:auto;left:0;position:absolute;width:72px}.app-article-masthead__journal-link{display:block;font-size:1.125rem;font-weight:700;margin:0 0 8px;max-width:400px;padding:0 0 0 88px;position:relative}.app-article-masthead__journal-title{-webkit-box-orient:vertical;-webkit-line-clamp:3;display:-webkit-box;overflow:hidden}.app-article-masthead__submission-link{align-items:center;display:flex;font-size:1rem;gap:4px 4px;margin:0 0 0 88px}.app-article-masthead__access{align-items:center;display:flex;flex-wrap:wrap;font-size:.875rem;font-weight:300;gap:4px 4px;margin:0}.app-article-masthead__buttons{display:flex;flex-flow:column wrap;gap:16px 16px}.app-article-masthead__access svg,.app-masthead--pastel .c-pdf-download .u-button--primary svg,.app-masthead--pastel .c-pdf-download .u-button--secondary svg,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary svg,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary svg{fill:currentcolor}.app-article-masthead a{color:#fff}.app-masthead--pastel .c-pdf-download .u-button--primary,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary{background-color:#025e8d;background-image:none;border:2px solid transparent;box-shadow:none;color:#fff;font-weight:700}.app-masthead--pastel .c-pdf-download .u-button--primary:visited,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:visited{color:#fff}.app-masthead--pastel .c-pdf-download .u-button--primary:hover,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:hover{text-decoration:none}.app-masthead--pastel .c-pdf-download .u-button--primary:focus,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:focus{border:4px solid #fc0;box-shadow:none;outline:0;text-decoration:none}.app-masthead--pastel .c-pdf-download .u-button--primary:focus,.app-masthead--pastel .c-pdf-download .u-button--primary:hover,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:focus,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:hover{background-color:#fff;background-image:none;color:#01324b}.app-masthead--pastel .c-pdf-download .u-button--primary:hover,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:hover{background:0 0;border:2px solid #025e8d;box-shadow:none;color:#025e8d}.app-masthead--pastel .c-pdf-download .u-button--secondary,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary{background:0 0;border:2px solid #025e8d;color:#025e8d;font-weight:700}.app-masthead--pastel .c-pdf-download .u-button--secondary:visited,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary:visited{color:#01324b}.app-masthead--pastel .c-pdf-download .u-button--secondary:hover,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary:hover{background-color:#01324b;background-color:#025e8d;border:2px solid transparent;box-shadow:none;color:#fff}.app-masthead--pastel .c-pdf-download .u-button--secondary:focus,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary:focus{background-color:#fff;background-image:none;border:4px solid #fc0;color:#01324b}@media only screen and (min-width:768px){.app-article-masthead{flex-direction:row;gap:64px 64px;padding:24px 0}.app-article-masthead__brand{border:0;padding:0}.app-article-masthead__brand img{height:auto;position:static;width:auto}.app-article-masthead__buttons{align-items:center;flex-direction:row;margin-top:auto}.app-article-masthead__journal-link{display:flex;flex-direction:column;gap:24px 24px;margin:0 0 8px;padding:0}.app-article-masthead__submission-link{margin:0}}@media only screen and (min-width:1024px){.app-article-masthead__brand{flex-basis:400px}}.app-article-masthead .c-article-identifiers{font-size:.875rem;font-weight:300;line-height:1;margin:0 0 8px;overflow:hidden;padding:0}.app-article-masthead .c-article-identifiers--cite-list{margin:0 0 16px}.app-article-masthead .c-article-identifiers *{color:#fff}.app-article-masthead .c-cod{display:none}.app-article-masthead .c-article-identifiers__item{border-left:1px solid #fff;border-right:0;margin:0 17px 8px -9px;padding:0 0 0 8px}.app-article-masthead .c-article-identifiers__item--cite{border-left:0}.app-article-metrics-bar{display:flex;flex-wrap:wrap;font-size:1rem;padding:16px 0 0;row-gap:24px}.app-article-metrics-bar__item{padding:0 16px 0 0}.app-article-metrics-bar__count{font-weight:700}.app-article-metrics-bar__label{font-weight:400;padding-left:4px}.app-article-metrics-bar__icon{height:auto;margin-right:4px;margin-top:-4px;width:auto}.app-article-metrics-bar__arrow-icon{margin:4px 0 0 4px}.app-article-metrics-bar a{color:#000}.app-article-metrics-bar .app-article-metrics-bar__item--metrics{padding-right:0}.app-overview-section .c-article-author-list,.app-overview-section__authors{line-height:2}.app-article-metrics-bar{margin-top:8px}.c-book-toc-pagination+.c-book-section__back-to-top{margin-top:0}.c-article-body .c-article-access-provider__text--chapter{color:#222;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;padding:20px 0}.c-article-body .c-article-access-provider__text--chapter svg.c-status-message__icon{fill:#003f8d;vertical-align:middle}.c-article-body-section__content--separator{padding-top:40px}.c-pdf-download__link{max-height:44px}.app-article-access .u-button--primary,.app-article-access .u-button--primary:visited{color:#fff}.c-article-sidebar{display:none}@media only screen and (min-width:1024px){.c-article-sidebar{display:block}}.c-cod__form{border-radius:12px}.c-cod__label{font-size:.875rem}.c-cod .c-status-message{align-items:center;justify-content:center;margin-bottom:16px;padding-bottom:16px}@media only screen and (min-width:1024px){.c-cod .c-status-message{align-items:inherit}}.c-cod .c-status-message__icon{margin-top:4px}.c-cod .c-cod__prompt{font-size:1rem;margin-bottom:16px}.c-article-body .app-article-access,.c-book-body .app-article-access{display:block}@media only screen and (min-width:1024px){.c-article-body .app-article-access,.c-book-body .app-article-access{display:none}}.c-article-body .app-card-service{margin-bottom:32px}@media only screen and (min-width:1024px){.c-article-body .app-card-service{display:none}}.app-article-access .buybox__buy .u-button--secondary,.app-article-access .u-button--primary,.c-cod__row .u-button--primary{background-color:#025e8d;border:2px solid #025e8d;box-shadow:none;font-size:1rem;font-weight:700;gap:8px 8px;justify-content:center;line-height:1.5;padding:8px 24px}.app-article-access .buybox__buy .u-button--secondary,.app-article-access .u-button--primary:hover,.c-cod__row .u-button--primary:hover{background-color:#fff;color:#025e8d}.app-article-access .buybox__buy .u-button--secondary:hover{background-color:#025e8d;color:#fff}.buybox__buy .c-notes__text{color:#666;font-size:.875rem;padding:0 16px 8px}.c-cod__input{flex-basis:auto;width:100%}.c-article-title{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:2.25rem;font-weight:700;line-height:1.2;margin:12px 0}.c-reading-companion__figure-item figure{margin:0}@media only screen and (min-width:768px){.c-article-title{margin:16px 0}}.app-article-access{border:1px solid #c5e0f4;border-radius:12px}.app-article-access__heading{border-bottom:1px solid #c5e0f4;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:1.125rem;font-weight:700;margin:0;padding:16px;text-align:center}.app-article-access .buybox__info svg{vertical-align:middle}.c-article-body .app-article-access p{margin-bottom:0}.app-article-access .buybox__info{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:1rem;margin:0}.app-article-access{margin:0 0 32px}@media only screen and (min-width:1024px){.app-article-access{margin:0 0 24px}}.c-status-message{font-size:1rem}.c-article-body{font-size:1.125rem}.c-article-body dl,.c-article-body ol,.c-article-body p,.c-article-body ul{margin-bottom:32px;margin-top:0}.c-article-access-provider__text:last-of-type,.c-article-body .c-notes__text:last-of-type{margin-bottom:0}.c-article-body ol p,.c-article-body ul p{margin-bottom:16px}.c-article-section__figure-caption{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.c-reading-companion__figure-item{border-top-color:#c5e0f4}.c-reading-companion__sticky{max-width:400px}.c-article-section .c-article-section__figure-description>*{font-size:1rem;margin-bottom:16px}.c-reading-companion__reference-item{border-top:1px solid #d5d5d5;padding:16px 0}.c-reading-companion__reference-item:first-child{padding-top:0}.c-article-share-box__button,.js .c-article-authors-search__item .c-article-button{background:0 0;border:2px solid #025e8d;border-radius:32px;box-shadow:none;color:#025e8d;font-size:1rem;font-weight:700;line-height:1.5;margin:0;padding:8px 24px;transition:all .2s ease 0s}.c-article-authors-search__item .c-article-button{width:100%}.c-pdf-download .u-button{background-color:#fff;border:2px solid #fff;color:#01324b;justify-content:center}.c-context-bar__container .c-pdf-download .u-button svg,.c-pdf-download .u-button svg{fill:currentcolor}.c-pdf-download .u-button:visited{color:#01324b}.c-pdf-download .u-button:hover{border:4px solid #01324b;box-shadow:none}.c-pdf-download .u-button:focus,.c-pdf-download .u-button:hover{background-color:#01324b}.c-pdf-download .u-button:focus svg path,.c-pdf-download .u-button:hover svg path{fill:#fff}.c-context-bar__container .c-pdf-download .u-button{background-image:none;border:2px solid;color:#fff}.c-context-bar__container .c-pdf-download .u-button:visited{color:#fff}.c-context-bar__container .c-pdf-download .u-button:hover{text-decoration:none}.c-context-bar__container .c-pdf-download .u-button:focus{box-shadow:none;outline:0;text-decoration:none}.c-context-bar__container .c-pdf-download .u-button:focus,.c-context-bar__container .c-pdf-download .u-button:hover{background-color:#fff;background-image:none;color:#01324b}.c-context-bar__container .c-pdf-download .u-button:focus svg path,.c-context-bar__container .c-pdf-download .u-button:hover svg path{fill:#01324b}.c-context-bar__container .c-pdf-download .u-button,.c-pdf-download .u-button{box-shadow:none;font-size:1rem;font-weight:700;line-height:1.5;padding:8px 24px}.c-context-bar__container .c-pdf-download .u-button{background-color:#025e8d}.c-pdf-download .u-button:hover{border:2px solid #fff}.c-pdf-download .u-button:focus,.c-pdf-download .u-button:hover{background:0 0;box-shadow:none;color:#fff}.c-context-bar__container .c-pdf-download .u-button:hover{border:2px solid #025e8d;box-shadow:none;color:#025e8d}.c-context-bar__container .c-pdf-download .u-button:focus,.c-pdf-download .u-button:focus{border:2px solid #025e8d}.c-article-share-box__button:focus:focus,.c-article__pill-button:focus:focus,.c-context-bar__container .c-pdf-download .u-button:focus:focus,.c-pdf-download .u-button:focus:focus{outline:3px solid #08c;will-change:transform}.c-pdf-download__link .u-icon{padding-top:0}.c-bibliographic-information__column button{margin-bottom:16px}.c-article-body .c-article-author-affiliation__list p,.c-article-body .c-article-author-information__list p,figure{margin:0}.c-article-share-box__button{margin-right:16px}.c-status-message--boxed{border-radius:12px}.c-article-associated-content__collection-title{font-size:1rem}.app-card-service__description,.c-article-body .app-card-service__description{color:#222;margin-bottom:0;margin-top:8px}.app-article-access__subscriptions a,.app-article-access__subscriptions a:visited,.app-book-series-listing__item a,.app-book-series-listing__item a:hover,.app-book-series-listing__item a:visited,.c-article-author-list a,.c-article-author-list a:visited,.c-article-buy-box a,.c-article-buy-box a:visited,.c-article-peer-review a,.c-article-peer-review a:visited,.c-article-satellite-subtitle a,.c-article-satellite-subtitle a:visited,.c-breadcrumbs__link,.c-breadcrumbs__link:hover,.c-breadcrumbs__link:visited{color:#000}.c-article-author-list svg{height:24px;margin:0 0 0 6px;width:24px}.c-article-header{margin-bottom:32px}@media only screen and (min-width:876px){.js .c-ad--conditional{display:block}}.u-lazy-ad-wrapper{background-color:#fff;display:none;min-height:149px}@media only screen and (min-width:876px){.u-lazy-ad-wrapper{display:block}}p.c-ad__label{margin-bottom:4px}.c-ad--728x90{background-color:#fff;border-bottom:2px solid #cedbe0} } </style> <style>@media only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark) { .eds-c-header__brand img{height:24px;width:203px}.app-article-masthead__journal-link img{height:93px;width:72px}@media only screen and (min-width:769px){.app-article-masthead__journal-link img{height:161px;width:122px}} } </style> <link rel="stylesheet" data-test="critical-css-handler" data-inline-css-source="critical-css" href=/oscar-static/app-springerlink/css/core-darwin-9fe647df8f.css media="print" onload="this.media='all';this.onload=null"> <link rel="stylesheet" data-test="critical-css-handler" data-inline-css-source="critical-css" href="/oscar-static/app-springerlink/css/enhanced-darwin-article-8aaaca8a1c.css" media="print" onload="this.media='only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark)';this.onload=null"> <script type="text/javascript"> config = { env: 'live', site: '40062.springer.com', siteWithPath: '40062.springer.com' + window.location.pathname, twitterHashtag: '40062', cmsPrefix: 'https://studio-cms.springernature.com/studio/', publisherBrand: 'Springer', mustardcut: false }; </script> <script> window.dataLayer = [{"GA Key":"UA-26408784-1","DOI":"10.1007/s40062-016-0139-x","Page":"article","springerJournal":true,"Publishing Model":"Hybrid Access","Country":"SG","japan":false,"doi":"10.1007-s40062-016-0139-x","Journal Id":40062,"Journal Title":"Journal of Homotopy and Related Structures","imprint":"Springer","Keywords":"Homotopy theory, Homotopy colimit, Quasicategory, Cofibration category","kwrd":["Homotopy_theory","Homotopy_colimit","Quasicategory","Cofibration_category"],"Labs":"Y","ksg":"Krux.segments","kuid":"Krux.uid","Has Body":"Y","Features":[],"Open Access":"N","hasAccess":"Y","bypassPaywall":"N","user":{"license":{"businessPartnerID":[],"businessPartnerIDString":""}},"Access Type":"permanently-free","Bpids":"","Bpnames":"","BPID":["1"],"VG Wort Identifier":"pw-vgzm.415900-10.1007-s40062-016-0139-x","Full HTML":"Y","Subject Codes":["SCM","SCM28019","SCM11000","SCM12066","SCM25001"],"pmc":["M","M28019","M11000","M12066","M25001"],"session":{"authentication":{"loginStatus":"N"},"attributes":{"edition":"academic"}},"content":{"serial":{"eissn":"1512-2891","pissn":"2193-8407"},"type":"Article","category":{"pmc":{"primarySubject":"Mathematics","primarySubjectCode":"M","secondarySubjects":{"1":"Algebraic Topology","2":"Algebra","3":"Functional Analysis","4":"Number Theory"},"secondarySubjectCodes":{"1":"M28019","2":"M11000","3":"M12066","4":"M25001"}},"sucode":"SC10","articleType":"Article"},"attributes":{"deliveryPlatform":"oscar"}},"page":{"attributes":{"environment":"live"},"category":{"pageType":"article"}},"Event Category":"Article"}]; </script> <script data-test="springer-link-article-datalayer"> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ ga4MeasurementId: 'G-B3E4QL2TPR', ga360TrackingId: 'UA-26408784-1', twitterId: 'o47a7', baiduId: 'aef3043f025ccf2305af8a194652d70b', ga4ServerUrl: 'https://collect.springer.com', imprint: 'springerlink', page: { attributes:{ featureFlags: [{ name: 'darwin-orion', active: true }, { name: 'chapter-books-recs', active: true } ], darwinAvailable: true } } }); </script> <script> (function(w, d) { w.config = w.config || {}; w.config.mustardcut = false; if (w.matchMedia && w.matchMedia('only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark)').matches) { w.config.mustardcut = true; d.classList.add('js'); d.classList.remove('grade-c'); d.classList.remove('no-js'); } })(window, document.documentElement); </script> <script class="js-entry"> if (window.config.mustardcut) { (function(w, d) { window.Component = {}; window.suppressShareButton = false; window.onArticlePage = true; var currentScript = d.currentScript || d.head.querySelector('script.js-entry'); function catchNoModuleSupport() { var scriptEl = d.createElement('script'); return (!('noModule' in scriptEl) && 'onbeforeload' in scriptEl) } var headScripts = [ {'src': '/oscar-static/js/polyfill-es5-bundle-572d4fec60.js', 'async': false} ]; var bodyScripts = [ {'src': '/oscar-static/js/global-article-es5-bundle-dad1690b0d.js', 'async': false, 'module': false}, {'src': '/oscar-static/js/global-article-es6-bundle-e7d03c4cb3.js', 'async': false, 'module': true} ]; function createScript(script) { var scriptEl = d.createElement('script'); scriptEl.src = script.src; scriptEl.async = script.async; if (script.module === true) { scriptEl.type = "module"; if (catchNoModuleSupport()) { scriptEl.src = ''; } } else if (script.module === false) { scriptEl.setAttribute('nomodule', true) } if (script.charset) { scriptEl.setAttribute('charset', script.charset); } return scriptEl; } for (var i = 0; i < headScripts.length; ++i) { var scriptEl = createScript(headScripts[i]); currentScript.parentNode.insertBefore(scriptEl, currentScript.nextSibling); } d.addEventListener('DOMContentLoaded', function() { for (var i = 0; i < bodyScripts.length; ++i) { var scriptEl = createScript(bodyScripts[i]); d.body.appendChild(scriptEl); } }); // Webfont repeat view var config = w.config; if (config && config.publisherBrand && sessionStorage.fontsLoaded === 'true') { d.documentElement.className += ' webfonts-loaded'; } })(window, document); } </script> <script data-src="https://cdn.optimizely.com/js/27195530232.js" data-cc-script="C03"></script> <script data-test="gtm-head"> window.initGTM = function() { if (window.config.mustardcut) { (function (w, d, s, l, i) { w[l] = w[l] || []; w[l].push({'gtm.start': new Date().getTime(), event: 'gtm.js'}); var f = d.getElementsByTagName(s)[0], j = d.createElement(s), dl = l != 'dataLayer' ? '&l=' + l : ''; j.async = true; j.src = 'https://www.googletagmanager.com/gtm.js?id=' + i + dl; f.parentNode.insertBefore(j, f); })(window, document, 'script', 'dataLayer', 'GTM-MRVXSHQ'); } } </script> <script> (function (w, d, t) { function cc() { var h = w.location.hostname; var e = d.createElement(t), s = d.getElementsByTagName(t)[0]; if (h.indexOf('springer.com') > -1 && h.indexOf('biomedcentral.com') === -1 && h.indexOf('springeropen.com') === -1) { if (h.indexOf('link-qa.springer.com') > -1 || h.indexOf('test-www.springer.com') > -1) { e.src = 'https://cmp.springer.com/production_live/en/consent-bundle-17-52.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } else { e.src = 'https://cmp.springer.com/production_live/en/consent-bundle-17-52.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } } else if (h.indexOf('biomedcentral.com') > -1) { if (h.indexOf('biomedcentral.com.qa') > -1) { e.src = 'https://cmp.biomedcentral.com/production_live/en/consent-bundle-15-38.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } else { e.src = 'https://cmp.biomedcentral.com/production_live/en/consent-bundle-15-38.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } } else if (h.indexOf('springeropen.com') > -1) { if (h.indexOf('springeropen.com.qa') > -1) { e.src = 'https://cmp.springernature.com/production_live/en/consent-bundle-16-35.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } else { e.src = 'https://cmp.springernature.com/production_live/en/consent-bundle-16-35.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } } else if (h.indexOf('springernature.com') > -1) { if (h.indexOf('beta-qa.springernature.com') > -1) { e.src = 'https://cmp.springernature.com/production_live/en/consent-bundle-49-43.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-NK22KLS')"); } else { e.src = 'https://cmp.springernature.com/production_live/en/consent-bundle-49-43.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-NK22KLS')"); } } else { e.src = '/oscar-static/js/cookie-consent-es5-bundle-cb57c2c98a.js'; e.setAttribute('data-consent', h); } s.insertAdjacentElement('afterend', e); } cc(); })(window, document, 'script'); </script> <link rel="canonical" href="https://link.springer.com/article/10.1007/s40062-016-0139-x"/> <script type="application/ld+json">{"mainEntity":{"headline":"Frames in cofibration categories","description":"We introduce the quasicategory of frames of a cofibration category, i.e. a new model of the \n \n \n \n $$(\\infty ,1)$$\n \n \n \n -category associated with a cofibration category.","datePublished":"2016-07-13T00:00:00Z","dateModified":"2016-07-13T00:00:00Z","pageStart":"577","pageEnd":"616","sameAs":"https://doi.org/10.1007/s40062-016-0139-x","keywords":["Homotopy theory","Homotopy colimit","Quasicategory","Cofibration category","Algebraic Topology","Algebra","Functional Analysis","Number Theory"],"image":["https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figg_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Fign_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figo_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figp_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figq_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figr_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figs_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figt_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figu_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figv_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figw_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figx_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figy_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figz_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figaa_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figab_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figac_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figad_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figae_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figaf_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figag_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figah_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figai_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figaj_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figak_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figan_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figao_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figap_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figaq_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figar_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figas_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figav_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figaw_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figax_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figay_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figaz_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figba_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbb_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbc_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbd_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbe_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbf_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbg_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbh_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbi_HTML.gif"],"isPartOf":{"name":"Journal of Homotopy and Related Structures","issn":["1512-2891","2193-8407"],"volumeNumber":"12","@type":["Periodical","PublicationVolume"]},"publisher":{"name":"Springer Berlin Heidelberg","logo":{"url":"https://www.springernature.com/app-sn/public/images/logo-springernature.png","@type":"ImageObject"},"@type":"Organization"},"author":[{"name":"Karol Szumiło","affiliation":[{"name":"University of Western Ontario","address":{"name":"Department of Mathematics, University of Western Ontario, London, Canada","@type":"PostalAddress"},"@type":"Organization"}],"email":"kszumilo@uwo.ca","@type":"Person"}],"isAccessibleForFree":true,"@type":"ScholarlyArticle"},"@context":"https://schema.org","@type":"WebPage"}</script> </head> <body class="" >  <noscript> <iframe src="https://www.googletagmanager.com/ns.html?id=GTM-MRVXSHQ" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript>   <noscript data-test="gtm-body"> <iframe src="https://www.googletagmanager.com/ns.html?id=GTM-MRVXSHQ" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript>  <div class="u-visually-hidden" aria-hidden="true" data-test="darwin-icons"> <?xml version="1.0" encoding="UTF-8"?><!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd"><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><symbol id="icon-eds-i-accesses-medium" viewBox="0 0 24 24"><path d="M15.59 1a1 1 0 0 1 .706.291l5.41 5.385a1 1 0 0 1 .294.709v13.077c0 .674-.269 1.32-.747 1.796a2.549 2.549 0 0 1-1.798.742H15a1 1 0 0 1 0-2h4.455a.549.549 0 0 0 .387-.16.535.535 0 0 0 .158-.378V7.8L15.178 3H5.545a.543.543 0 0 0-.538.451L5 3.538v8.607a1 1 0 0 1-2 0V3.538A2.542 2.542 0 0 1 5.545 1h10.046ZM8 13c2.052 0 4.66 1.61 6.36 3.4l.124.141c.333.41.516.925.516 1.459 0 .6-.232 1.178-.64 1.599C12.666 21.388 10.054 23 8 23c-2.052 0-4.66-1.61-6.353-3.393A2.31 2.31 0 0 1 1 18c0-.6.232-1.178.64-1.6C3.34 14.61 5.948 13 8 13Zm0 2c-1.369 0-3.552 1.348-4.917 2.785A.31.31 0 0 0 3 18c0 .083.031.161.09.222C4.447 19.652 6.631 21 8 21c1.37 0 3.556-1.35 4.917-2.785A.31.31 0 0 0 13 18a.32.32 0 0 0-.048-.17l-.042-.052C11.553 16.348 9.369 15 8 15Zm0 1a2 2 0 1 1 0 4 2 2 0 0 1 0-4Z"/></symbol><symbol id="icon-eds-i-altmetric-medium" viewBox="0 0 24 24"><path d="M12 1c5.978 0 10.843 4.77 10.996 10.712l.004.306-.002.022-.002.248C22.843 18.23 17.978 23 12 23 5.925 23 1 18.075 1 12S5.925 1 12 1Zm-1.726 9.246L8.848 12.53a1 1 0 0 1-.718.461L8.003 13l-4.947.014a9.001 9.001 0 0 0 17.887-.001L16.553 13l-2.205 3.53a1 1 0 0 1-1.735-.068l-.05-.11-2.289-6.106ZM12 3a9.001 9.001 0 0 0-8.947 8.013l4.391-.012L9.652 7.47a1 1 0 0 1 1.784.179l2.288 6.104 1.428-2.283a1 1 0 0 1 .722-.462l.129-.008 4.943.012A9.001 9.001 0 0 0 12 3Z"/></symbol><symbol id="icon-eds-i-arrow-bend-down-medium" viewBox="0 0 24 24"><path d="m11.852 20.989.058.007L12 21l.075-.003.126-.017.111-.03.111-.044.098-.052.104-.074.082-.073 6-6a1 1 0 0 0-1.414-1.414L13 17.585v-12.2C13 4.075 11.964 3 10.667 3H4a1 1 0 1 0 0 2h6.667c.175 0 .333.164.333.385v12.2l-4.293-4.292a1 1 0 0 0-1.32-.083l-.094.083a1 1 0 0 0 0 1.414l6 6c.035.036.073.068.112.097l.11.071.114.054.105.035.118.025Z"/></symbol><symbol id="icon-eds-i-arrow-bend-down-small" viewBox="0 0 16 16"><path d="M1 2a1 1 0 0 0 1 1h5v8.585L3.707 8.293a1 1 0 0 0-1.32-.083l-.094.083a1 1 0 0 0 0 1.414l5 5 .063.059.093.069.081.048.105.048.104.035.105.022.096.01h.136l.122-.018.113-.03.103-.04.1-.053.102-.07.052-.043 5.04-5.037a1 1 0 1 0-1.415-1.414L9 11.583V3a2 2 0 0 0-2-2H2a1 1 0 0 0-1 1Z"/></symbol><symbol id="icon-eds-i-arrow-bend-up-medium" viewBox="0 0 24 24"><path d="m11.852 3.011.058-.007L12 3l.075.003.126.017.111.03.111.044.098.052.104.074.082.073 6 6a1 1 0 1 1-1.414 1.414L13 6.415v12.2C13 19.925 11.964 21 10.667 21H4a1 1 0 0 1 0-2h6.667c.175 0 .333-.164.333-.385v-12.2l-4.293 4.292a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414l6-6c.035-.036.073-.068.112-.097l.11-.071.114-.054.105-.035.118-.025Z"/></symbol><symbol id="icon-eds-i-arrow-bend-up-small" viewBox="0 0 16 16"><path d="M1 13.998a1 1 0 0 1 1-1h5V4.413L3.707 7.705a1 1 0 0 1-1.32.084l-.094-.084a1 1 0 0 1 0-1.414l5-5 .063-.059.093-.068.081-.05.105-.047.104-.035.105-.022L7.94 1l.136.001.122.017.113.03.103.04.1.053.102.07.052.043 5.04 5.037a1 1 0 1 1-1.415 1.414L9 4.415v8.583a2 2 0 0 1-2 2H2a1 1 0 0 1-1-1Z"/></symbol><symbol id="icon-eds-i-arrow-diagonal-medium" viewBox="0 0 24 24"><path d="M14 3h6l.075.003.126.017.111.03.111.044.098.052.096.067.09.08c.036.035.068.073.097.112l.071.11.054.114.035.105.03.148L21 4v6a1 1 0 0 1-2 0V6.414l-4.293 4.293a1 1 0 0 1-1.414-1.414L17.584 5H14a1 1 0 0 1-.993-.883L13 4a1 1 0 0 1 1-1ZM4 13a1 1 0 0 1 1 1v3.584l4.293-4.291a1 1 0 1 1 1.414 1.414L6.414 19H10a1 1 0 0 1 .993.883L11 20a1 1 0 0 1-1 1l-6.075-.003-.126-.017-.111-.03-.111-.044-.098-.052-.096-.067-.09-.08a1.01 1.01 0 0 1-.097-.112l-.071-.11-.054-.114-.035-.105-.025-.118-.007-.058L3 20v-6a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-arrow-diagonal-small" viewBox="0 0 16 16"><path d="m2 15-.082-.004-.119-.016-.111-.03-.111-.044-.098-.052-.096-.067-.09-.08a1.008 1.008 0 0 1-.097-.112l-.071-.11-.031-.062-.034-.081-.024-.076-.025-.118-.007-.058L1 14.02V9a1 1 0 1 1 2 0v2.584l2.793-2.791a1 1 0 1 1 1.414 1.414L4.414 13H7a1 1 0 0 1 .993.883L8 14a1 1 0 0 1-1 1H2ZM14 1l.081.003.12.017.111.03.111.044.098.052.096.067.09.08c.036.035.068.073.097.112l.071.11.031.062.034.081.024.076.03.148L15 2v5a1 1 0 0 1-2 0V4.414l-2.96 2.96A1 1 0 1 1 8.626 5.96L11.584 3H9a1 1 0 0 1-.993-.883L8 2a1 1 0 0 1 1-1h5Z"/></symbol><symbol id="icon-eds-i-arrow-down-medium" viewBox="0 0 24 24"><path d="m20.707 12.728-7.99 7.98a.996.996 0 0 1-.561.281l-.157.011a.998.998 0 0 1-.788-.384l-7.918-7.908a1 1 0 0 1 1.414-1.416L11 17.576V4a1 1 0 0 1 2 0v13.598l6.293-6.285a1 1 0 0 1 1.32-.082l.095.083a1 1 0 0 1-.001 1.414Z"/></symbol><symbol id="icon-eds-i-arrow-down-small" viewBox="0 0 16 16"><path d="m1.293 8.707 6 6 .063.059.093.069.081.048.105.049.104.034.056.013.118.017L8 15l.076-.003.122-.017.113-.03.085-.032.063-.03.098-.058.06-.043.05-.043 6.04-6.037a1 1 0 0 0-1.414-1.414L9 11.583V2a1 1 0 1 0-2 0v9.585L2.707 7.293a1 1 0 0 0-1.32-.083l-.094.083a1 1 0 0 0 0 1.414Z"/></symbol><symbol id="icon-eds-i-arrow-left-medium" viewBox="0 0 24 24"><path d="m11.272 3.293-7.98 7.99a.996.996 0 0 0-.281.561L3 12.001c0 .32.15.605.384.788l7.908 7.918a1 1 0 0 0 1.416-1.414L6.424 13H20a1 1 0 0 0 0-2H6.402l6.285-6.293a1 1 0 0 0 .082-1.32l-.083-.095a1 1 0 0 0-1.414.001Z"/></symbol><symbol id="icon-eds-i-arrow-left-small" viewBox="0 0 16 16"><path d="m7.293 1.293-6 6-.059.063-.069.093-.048.081-.049.105-.034.104-.013.056-.017.118L1 8l.003.076.017.122.03.113.032.085.03.063.058.098.043.06.043.05 6.037 6.04a1 1 0 0 0 1.414-1.414L4.417 9H14a1 1 0 0 0 0-2H4.415l4.292-4.293a1 1 0 0 0 .083-1.32l-.083-.094a1 1 0 0 0-1.414 0Z"/></symbol><symbol id="icon-eds-i-arrow-right-medium" viewBox="0 0 24 24"><path d="m12.728 3.293 7.98 7.99a.996.996 0 0 1 .281.561l.011.157c0 .32-.15.605-.384.788l-7.908 7.918a1 1 0 0 1-1.416-1.414L17.576 13H4a1 1 0 0 1 0-2h13.598l-6.285-6.293a1 1 0 0 1-.082-1.32l.083-.095a1 1 0 0 1 1.414.001Z"/></symbol><symbol id="icon-eds-i-arrow-right-small" viewBox="0 0 16 16"><path d="m8.707 1.293 6 6 .059.063.069.093.048.081.049.105.034.104.013.056.017.118L15 8l-.003.076-.017.122-.03.113-.032.085-.03.063-.058.098-.043.06-.043.05-6.037 6.04a1 1 0 0 1-1.414-1.414L11.583 9H2a1 1 0 1 1 0-2h9.585L7.293 2.707a1 1 0 0 1-.083-1.32l.083-.094a1 1 0 0 1 1.414 0Z"/></symbol><symbol id="icon-eds-i-arrow-up-medium" viewBox="0 0 24 24"><path d="m3.293 11.272 7.99-7.98a.996.996 0 0 1 .561-.281L12.001 3c.32 0 .605.15.788.384l7.918 7.908a1 1 0 0 1-1.414 1.416L13 6.424V20a1 1 0 0 1-2 0V6.402l-6.293 6.285a1 1 0 0 1-1.32.082l-.095-.083a1 1 0 0 1 .001-1.414Z"/></symbol><symbol id="icon-eds-i-arrow-up-small" viewBox="0 0 16 16"><path d="m1.293 7.293 6-6 .063-.059.093-.069.081-.048.105-.049.104-.034.056-.013.118-.017L8 1l.076.003.122.017.113.03.085.032.063.03.098.058.06.043.05.043 6.04 6.037a1 1 0 0 1-1.414 1.414L9 4.417V14a1 1 0 0 1-2 0V4.415L2.707 8.707a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414Z"/></symbol><symbol id="icon-eds-i-article-medium" viewBox="0 0 24 24"><path d="M8 7a1 1 0 0 0 0 2h4a1 1 0 1 0 0-2H8ZM8 11a1 1 0 1 0 0 2h8a1 1 0 1 0 0-2H8ZM7 16a1 1 0 0 1 1-1h8a1 1 0 1 1 0 2H8a1 1 0 0 1-1-1Z"/><path d="M5.545 1A2.542 2.542 0 0 0 3 3.538v16.924A2.542 2.542 0 0 0 5.545 23h12.91A2.542 2.542 0 0 0 21 20.462V3.5A2.5 2.5 0 0 0 18.5 1H5.545ZM5 3.538C5 3.245 5.24 3 5.545 3H18.5a.5.5 0 0 1 .5.5v16.962c0 .293-.24.538-.546.538H5.545A.542.542 0 0 1 5 20.462V3.538Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-book-medium" viewBox="0 0 24 24"><path d="M18.5 1A2.5 2.5 0 0 1 21 3.5v12c0 1.16-.79 2.135-1.86 2.418l-.14.031V21h1a1 1 0 0 1 .993.883L21 22a1 1 0 0 1-1 1H6.5A3.5 3.5 0 0 1 3 19.5v-15A3.5 3.5 0 0 1 6.5 1h12ZM17 18H6.5a1.5 1.5 0 0 0-1.493 1.356L5 19.5A1.5 1.5 0 0 0 6.5 21H17v-3Zm1.5-15h-12A1.5 1.5 0 0 0 5 4.5v11.837l.054-.025a3.481 3.481 0 0 1 1.254-.307L6.5 16h12a.5.5 0 0 0 .492-.41L19 15.5v-12a.5.5 0 0 0-.5-.5ZM15 6a1 1 0 0 1 0 2H9a1 1 0 1 1 0-2h6Z"/></symbol><symbol id="icon-eds-i-book-series-medium" viewBox="0 0 24 24"><path fill-rule="evenodd" d="M1 3.786C1 2.759 1.857 2 2.82 2H6.18c.964 0 1.82.759 1.82 1.786V4h3.168c.668 0 1.298.364 1.616.938.158-.109.333-.195.523-.252l3.216-.965c.923-.277 1.962.204 2.257 1.187l4.146 13.82c.296.984-.307 1.957-1.23 2.234l-3.217.965c-.923.277-1.962-.203-2.257-1.187L13 10.005v10.21c0 1.04-.878 1.785-1.834 1.785H7.833c-.291 0-.575-.07-.83-.195A1.849 1.849 0 0 1 6.18 22H2.821C1.857 22 1 21.241 1 20.214V3.786ZM3 4v11h3V4H3Zm0 16v-3h3v3H3Zm15.075-.04-.814-2.712 2.874-.862.813 2.712-2.873.862Zm1.485-5.49-2.874.862-2.634-8.782 2.873-.862 2.635 8.782ZM8 20V6h3v14H8Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-calendar-acceptance-medium" viewBox="0 0 24 24"><path d="M17 2a1 1 0 0 1 1 1v1h1.5C20.817 4 22 5.183 22 6.5v13c0 1.317-1.183 2.5-2.5 2.5h-15C3.183 22 2 20.817 2 19.5v-13C2 5.183 3.183 4 4.5 4a1 1 0 1 1 0 2c-.212 0-.5.288-.5.5v13c0 .212.288.5.5.5h15c.212 0 .5-.288.5-.5v-13c0-.212-.288-.5-.5-.5H18v1a1 1 0 0 1-2 0V3a1 1 0 0 1 1-1Zm-.534 7.747a1 1 0 0 1 .094 1.412l-4.846 5.538a1 1 0 0 1-1.352.141l-2.77-2.076a1 1 0 0 1 1.2-1.6l2.027 1.519 4.236-4.84a1 1 0 0 1 1.411-.094ZM7.5 2a1 1 0 0 1 1 1v1H14a1 1 0 0 1 0 2H8.5v1a1 1 0 1 1-2 0V3a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-calendar-date-medium" viewBox="0 0 24 24"><path d="M17 2a1 1 0 0 1 1 1v1h1.5C20.817 4 22 5.183 22 6.5v13c0 1.317-1.183 2.5-2.5 2.5h-15C3.183 22 2 20.817 2 19.5v-13C2 5.183 3.183 4 4.5 4a1 1 0 1 1 0 2c-.212 0-.5.288-.5.5v13c0 .212.288.5.5.5h15c.212 0 .5-.288.5-.5v-13c0-.212-.288-.5-.5-.5H18v1a1 1 0 0 1-2 0V3a1 1 0 0 1 1-1ZM8 15a1 1 0 1 1 0 2 1 1 0 0 1 0-2Zm4 0a1 1 0 1 1 0 2 1 1 0 0 1 0-2Zm-4-4a1 1 0 1 1 0 2 1 1 0 0 1 0-2Zm4 0a1 1 0 1 1 0 2 1 1 0 0 1 0-2Zm4 0a1 1 0 1 1 0 2 1 1 0 0 1 0-2ZM7.5 2a1 1 0 0 1 1 1v1H14a1 1 0 0 1 0 2H8.5v1a1 1 0 1 1-2 0V3a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-calendar-decision-medium" viewBox="0 0 24 24"><path d="M17 2a1 1 0 0 1 1 1v1h1.5C20.817 4 22 5.183 22 6.5v13c0 1.317-1.183 2.5-2.5 2.5h-15C3.183 22 2 20.817 2 19.5v-13C2 5.183 3.183 4 4.5 4a1 1 0 1 1 0 2c-.212 0-.5.288-.5.5v13c0 .212.288.5.5.5h15c.212 0 .5-.288.5-.5v-13c0-.212-.288-.5-.5-.5H18v1a1 1 0 0 1-2 0V3a1 1 0 0 1 1-1Zm-2.935 8.246 2.686 2.645c.34.335.34.883 0 1.218l-2.686 2.645a.858.858 0 0 1-1.213-.009.854.854 0 0 1 .009-1.21l1.05-1.035H7.984a.992.992 0 0 1-.984-1c0-.552.44-1 .984-1h5.928l-1.051-1.036a.854.854 0 0 1-.085-1.121l.076-.088a.858.858 0 0 1 1.213-.009ZM7.5 2a1 1 0 0 1 1 1v1H14a1 1 0 0 1 0 2H8.5v1a1 1 0 1 1-2 0V3a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-calendar-impact-factor-medium" viewBox="0 0 24 24"><path d="M17 2a1 1 0 0 1 1 1v1h1.5C20.817 4 22 5.183 22 6.5v13c0 1.317-1.183 2.5-2.5 2.5h-15C3.183 22 2 20.817 2 19.5v-13C2 5.183 3.183 4 4.5 4a1 1 0 1 1 0 2c-.212 0-.5.288-.5.5v13c0 .212.288.5.5.5h15c.212 0 .5-.288.5-.5v-13c0-.212-.288-.5-.5-.5H18v1a1 1 0 0 1-2 0V3a1 1 0 0 1 1-1Zm-3.2 6.924a.48.48 0 0 1 .125.544l-1.52 3.283h2.304c.27 0 .491.215.491.483a.477.477 0 0 1-.13.327l-4.18 4.484a.498.498 0 0 1-.69.031.48.48 0 0 1-.125-.544l1.52-3.284H9.291a.487.487 0 0 1-.491-.482c0-.121.047-.238.13-.327l4.18-4.484a.498.498 0 0 1 .69-.031ZM7.5 2a1 1 0 0 1 1 1v1H14a1 1 0 0 1 0 2H8.5v1a1 1 0 1 1-2 0V3a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-call-papers-medium" viewBox="0 0 24 24"><g><path d="m20.707 2.883-1.414 1.414a1 1 0 0 0 1.414 1.414l1.414-1.414a1 1 0 0 0-1.414-1.414Z"/><path d="M6 16.054c0 2.026 1.052 2.943 3 2.943a1 1 0 1 1 0 2c-2.996 0-5-1.746-5-4.943v-1.227a4.068 4.068 0 0 1-1.83-1.189 4.553 4.553 0 0 1-.87-1.455 4.868 4.868 0 0 1-.3-1.686c0-1.17.417-2.298 1.17-3.14.38-.426.834-.767 1.338-1 .51-.237 1.06-.36 1.617-.36L6.632 6H7l7.932-2.895A2.363 2.363 0 0 1 18 5.36v9.28a2.36 2.36 0 0 1-3.069 2.25l.084.03L7 14.997H6v1.057Zm9.637-11.057a.415.415 0 0 0-.083.008L8 7.638v5.536l7.424 1.786.104.02c.035.01.072.02.109.02.2 0 .363-.16.363-.36V5.36c0-.2-.163-.363-.363-.363Zm-9.638 3h-.874a1.82 1.82 0 0 0-.625.111l-.15.063a2.128 2.128 0 0 0-.689.517c-.42.47-.661 1.123-.661 1.81 0 .34.06.678.176.992.114.308.28.585.485.816.4.447.925.691 1.464.691h.874v-5Z" clip-rule="evenodd"/><path d="M20 8.997h2a1 1 0 1 1 0 2h-2a1 1 0 1 1 0-2ZM20.707 14.293l1.414 1.414a1 1 0 0 1-1.414 1.414l-1.414-1.414a1 1 0 0 1 1.414-1.414Z"/></g></symbol><symbol id="icon-eds-i-card-medium" viewBox="0 0 24 24"><path d="M19.615 2c.315 0 .716.067 1.14.279.76.38 1.245 1.107 1.245 2.106v15.23c0 .315-.067.716-.279 1.14-.38.76-1.107 1.245-2.106 1.245H4.385a2.56 2.56 0 0 1-1.14-.279C2.485 21.341 2 20.614 2 19.615V4.385c0-.315.067-.716.279-1.14C2.659 2.485 3.386 2 4.385 2h15.23Zm0 2H4.385c-.213 0-.265.034-.317.14A.71.71 0 0 0 4 4.385v15.23c0 .213.034.265.14.317a.71.71 0 0 0 .245.068h15.23c.213 0 .265-.034.317-.14a.71.71 0 0 0 .068-.245V4.385c0-.213-.034-.265-.14-.317A.71.71 0 0 0 19.615 4ZM17 16a1 1 0 0 1 0 2H7a1 1 0 0 1 0-2h10Zm0-3a1 1 0 0 1 0 2H7a1 1 0 0 1 0-2h10Zm-.5-7A1.5 1.5 0 0 1 18 7.5v3a1.5 1.5 0 0 1-1.5 1.5h-9A1.5 1.5 0 0 1 6 10.5v-3A1.5 1.5 0 0 1 7.5 6h9ZM16 8H8v2h8V8Z"/></symbol><symbol id="icon-eds-i-cart-medium" viewBox="0 0 24 24"><path d="M5.76 1a1 1 0 0 1 .994.902L7.155 6h13.34c.18 0 .358.02.532.057l.174.045a2.5 2.5 0 0 1 1.693 3.103l-2.069 7.03c-.36 1.099-1.398 1.823-2.49 1.763H8.65c-1.272.015-2.352-.927-2.546-2.244L4.852 3H2a1 1 0 0 1-.993-.883L1 2a1 1 0 0 1 1-1h3.76Zm2.328 14.51a.555.555 0 0 0 .55.488l9.751.001a.533.533 0 0 0 .527-.357l2.059-7a.5.5 0 0 0-.48-.642H7.351l.737 7.51ZM18 19a2 2 0 1 1 0 4 2 2 0 0 1 0-4ZM8 19a2 2 0 1 1 0 4 2 2 0 0 1 0-4Z"/></symbol><symbol id="icon-eds-i-check-circle-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 2a9 9 0 1 0 0 18 9 9 0 0 0 0-18Zm5.125 4.72a1 1 0 0 1 .156 1.405l-6 7.5a1 1 0 0 1-1.421.143l-3-2.5a1 1 0 0 1 1.28-1.536l2.217 1.846 5.362-6.703a1 1 0 0 1 1.406-.156Z"/></symbol><symbol id="icon-eds-i-check-filled-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm5.125 6.72a1 1 0 0 0-1.406.155l-5.362 6.703-2.217-1.846a1 1 0 1 0-1.28 1.536l3 2.5a1 1 0 0 0 1.42-.143l6-7.5a1 1 0 0 0-.155-1.406Z"/></symbol><symbol id="icon-eds-i-chevron-down-medium" viewBox="0 0 24 24"><path d="M3.305 8.28a1 1 0 0 0-.024 1.415l7.495 7.762c.314.345.757.543 1.224.543.467 0 .91-.198 1.204-.522l7.515-7.783a1 1 0 1 0-1.438-1.39L12 15.845l-7.28-7.54A1 1 0 0 0 3.4 8.2l-.096.082Z"/></symbol><symbol id="icon-eds-i-chevron-down-small" viewBox="0 0 16 16"><path d="M13.692 5.278a1 1 0 0 1 .03 1.414L9.103 11.51a1.491 1.491 0 0 1-2.188.019L2.278 6.692a1 1 0 0 1 1.444-1.384L8 9.771l4.278-4.463a1 1 0 0 1 1.318-.111l.096.081Z"/></symbol><symbol id="icon-eds-i-chevron-left-medium" viewBox="0 0 24 24"><path d="M15.72 3.305a1 1 0 0 0-1.415-.024l-7.762 7.495A1.655 1.655 0 0 0 6 12c0 .467.198.91.522 1.204l7.783 7.515a1 1 0 1 0 1.39-1.438L8.155 12l7.54-7.28A1 1 0 0 0 15.8 3.4l-.082-.096Z"/></symbol><symbol id="icon-eds-i-chevron-left-small" viewBox="0 0 16 16"><path d="M10.722 2.308a1 1 0 0 0-1.414-.03L4.49 6.897a1.491 1.491 0 0 0-.019 2.188l4.838 4.637a1 1 0 1 0 1.384-1.444L6.229 8l4.463-4.278a1 1 0 0 0 .111-1.318l-.081-.096Z"/></symbol><symbol id="icon-eds-i-chevron-right-medium" viewBox="0 0 24 24"><path d="M8.28 3.305a1 1 0 0 1 1.415-.024l7.762 7.495c.345.314.543.757.543 1.224 0 .467-.198.91-.522 1.204l-7.783 7.515a1 1 0 1 1-1.39-1.438L15.845 12l-7.54-7.28A1 1 0 0 1 8.2 3.4l.082-.096Z"/></symbol><symbol id="icon-eds-i-chevron-right-small" viewBox="0 0 16 16"><path d="M5.278 2.308a1 1 0 0 1 1.414-.03l4.819 4.619a1.491 1.491 0 0 1 .019 2.188l-4.838 4.637a1 1 0 1 1-1.384-1.444L9.771 8 5.308 3.722a1 1 0 0 1-.111-1.318l.081-.096Z"/></symbol><symbol id="icon-eds-i-chevron-up-medium" viewBox="0 0 24 24"><path d="M20.695 15.72a1 1 0 0 0 .024-1.415l-7.495-7.762A1.655 1.655 0 0 0 12 6c-.467 0-.91.198-1.204.522l-7.515 7.783a1 1 0 1 0 1.438 1.39L12 8.155l7.28 7.54a1 1 0 0 0 1.319.106l.096-.082Z"/></symbol><symbol id="icon-eds-i-chevron-up-small" viewBox="0 0 16 16"><path d="M13.692 10.722a1 1 0 0 0 .03-1.414L9.103 4.49a1.491 1.491 0 0 0-2.188-.019L2.278 9.308a1 1 0 0 0 1.444 1.384L8 6.229l4.278 4.463a1 1 0 0 0 1.318.111l.096-.081Z"/></symbol><symbol id="icon-eds-i-citations-medium" viewBox="0 0 24 24"><path d="M15.59 1a1 1 0 0 1 .706.291l5.41 5.385a1 1 0 0 1 .294.709v13.077c0 .674-.269 1.32-.747 1.796a2.549 2.549 0 0 1-1.798.742h-5.843a1 1 0 1 1 0-2h5.843a.549.549 0 0 0 .387-.16.535.535 0 0 0 .158-.378V7.8L15.178 3H5.545a.543.543 0 0 0-.538.451L5 3.538v8.607a1 1 0 0 1-2 0V3.538A2.542 2.542 0 0 1 5.545 1h10.046ZM5.483 14.35c.197.26.17.62-.049.848l-.095.083-.016.011c-.36.24-.628.45-.804.634-.393.409-.59.93-.59 1.562.077-.019.192-.028.345-.028.442 0 .84.158 1.195.474.355.316.532.716.532 1.2 0 .501-.173.9-.518 1.198-.345.298-.767.446-1.266.446-.672 0-1.209-.195-1.612-.585-.403-.39-.604-.976-.604-1.757 0-.744.11-1.39.33-1.938.222-.549.49-1.009.807-1.38a4.28 4.28 0 0 1 .992-.88c.07-.043.148-.087.232-.133a.881.881 0 0 1 1.121.245Zm5 0c.197.26.17.62-.049.848l-.095.083-.016.011c-.36.24-.628.45-.804.634-.393.409-.59.93-.59 1.562.077-.019.192-.028.345-.028.442 0 .84.158 1.195.474.355.316.532.716.532 1.2 0 .501-.173.9-.518 1.198-.345.298-.767.446-1.266.446-.672 0-1.209-.195-1.612-.585-.403-.39-.604-.976-.604-1.757 0-.744.11-1.39.33-1.938.222-.549.49-1.009.807-1.38a4.28 4.28 0 0 1 .992-.88c.07-.043.148-.087.232-.133a.881.881 0 0 1 1.121.245Z"/></symbol><symbol id="icon-eds-i-clipboard-check-medium" viewBox="0 0 24 24"><path d="M14.4 1c1.238 0 2.274.865 2.536 2.024L18.5 3C19.886 3 21 4.14 21 5.535v14.93C21 21.86 19.886 23 18.5 23h-13C4.114 23 3 21.86 3 20.465V5.535C3 4.14 4.114 3 5.5 3h1.57c.27-1.147 1.3-2 2.53-2h4.8Zm4.115 4-1.59.024A2.601 2.601 0 0 1 14.4 7H9.6c-1.23 0-2.26-.853-2.53-2H5.5c-.27 0-.5.234-.5.535v14.93c0 .3.23.535.5.535h13c.27 0 .5-.234.5-.535V5.535c0-.3-.23-.535-.485-.535Zm-1.909 4.205a1 1 0 0 1 .19 1.401l-5.334 7a1 1 0 0 1-1.344.23l-2.667-1.75a1 1 0 1 1 1.098-1.672l1.887 1.238 4.769-6.258a1 1 0 0 1 1.401-.19ZM14.4 3H9.6a.6.6 0 0 0-.6.6v.8a.6.6 0 0 0 .6.6h4.8a.6.6 0 0 0 .6-.6v-.8a.6.6 0 0 0-.6-.6Z"/></symbol><symbol id="icon-eds-i-clipboard-report-medium" viewBox="0 0 24 24"><path d="M14.4 1c1.238 0 2.274.865 2.536 2.024L18.5 3C19.886 3 21 4.14 21 5.535v14.93C21 21.86 19.886 23 18.5 23h-13C4.114 23 3 21.86 3 20.465V5.535C3 4.14 4.114 3 5.5 3h1.57c.27-1.147 1.3-2 2.53-2h4.8Zm4.115 4-1.59.024A2.601 2.601 0 0 1 14.4 7H9.6c-1.23 0-2.26-.853-2.53-2H5.5c-.27 0-.5.234-.5.535v14.93c0 .3.23.535.5.535h13c.27 0 .5-.234.5-.535V5.535c0-.3-.23-.535-.485-.535Zm-2.658 10.929a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h7.857Zm0-3.929a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h7.857ZM14.4 3H9.6a.6.6 0 0 0-.6.6v.8a.6.6 0 0 0 .6.6h4.8a.6.6 0 0 0 .6-.6v-.8a.6.6 0 0 0-.6-.6Z"/></symbol><symbol id="icon-eds-i-close-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 2a9 9 0 1 0 0 18 9 9 0 0 0 0-18ZM8.707 7.293 12 10.585l3.293-3.292a1 1 0 0 1 1.414 1.414L13.415 12l3.292 3.293a1 1 0 0 1-1.414 1.414L12 13.415l-3.293 3.292a1 1 0 1 1-1.414-1.414L10.585 12 7.293 8.707a1 1 0 0 1 1.414-1.414Z"/></symbol><symbol id="icon-eds-i-cloud-upload-medium" viewBox="0 0 24 24"><path d="m12.852 10.011.028-.004L13 10l.075.003.126.017.086.022.136.052.098.052.104.074.082.073 3 3a1 1 0 0 1 0 1.414l-.094.083a1 1 0 0 1-1.32-.083L14 13.416V20a1 1 0 0 1-2 0v-6.586l-1.293 1.293a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414l3-3 .112-.097.11-.071.114-.054.105-.035.118-.025Zm.587-7.962c3.065.362 5.497 2.662 5.992 5.562l.013.085.207.073c2.117.782 3.496 2.845 3.337 5.097l-.022.226c-.297 2.561-2.503 4.491-5.124 4.502a1 1 0 1 1-.009-2c1.619-.007 2.967-1.186 3.147-2.733.179-1.542-.86-2.979-2.487-3.353-.512-.149-.894-.579-.981-1.165-.21-2.237-2-4.035-4.308-4.308-2.31-.273-4.497 1.06-5.25 3.19l-.049.113c-.234.468-.718.756-1.176.743-1.418.057-2.689.857-3.32 2.084a3.668 3.668 0 0 0 .262 3.798c.796 1.136 2.169 1.764 3.583 1.635a1 1 0 1 1 .182 1.992c-2.125.194-4.193-.753-5.403-2.48a5.668 5.668 0 0 1-.403-5.86c.85-1.652 2.449-2.79 4.323-3.092l.287-.039.013-.028c1.207-2.741 4.125-4.404 7.186-4.042Z"/></symbol><symbol id="icon-eds-i-collection-medium" viewBox="0 0 24 24"><path d="M21 7a1 1 0 0 1 1 1v12.5a2.5 2.5 0 0 1-2.5 2.5H8a1 1 0 0 1 0-2h11.5a.5.5 0 0 0 .5-.5V8a1 1 0 0 1 1-1Zm-5.5-5A2.5 2.5 0 0 1 18 4.5v12a2.5 2.5 0 0 1-2.5 2.5h-11A2.5 2.5 0 0 1 2 16.5v-12A2.5 2.5 0 0 1 4.5 2h11Zm0 2h-11a.5.5 0 0 0-.5.5v12a.5.5 0 0 0 .5.5h11a.5.5 0 0 0 .5-.5v-12a.5.5 0 0 0-.5-.5ZM13 13a1 1 0 0 1 0 2H7a1 1 0 0 1 0-2h6Zm0-3.5a1 1 0 0 1 0 2H7a1 1 0 0 1 0-2h6ZM13 6a1 1 0 0 1 0 2H7a1 1 0 1 1 0-2h6Z"/></symbol><symbol id="icon-eds-i-conference-series-medium" viewBox="0 0 24 24"><path fill-rule="evenodd" d="M4.5 2A2.5 2.5 0 0 0 2 4.5v11A2.5 2.5 0 0 0 4.5 18h2.37l-2.534 2.253a1 1 0 0 0 1.328 1.494L9.88 18H11v3a1 1 0 1 0 2 0v-3h1.12l4.216 3.747a1 1 0 0 0 1.328-1.494L17.13 18h2.37a2.5 2.5 0 0 0 2.5-2.5v-11A2.5 2.5 0 0 0 19.5 2h-15ZM20 6V4.5a.5.5 0 0 0-.5-.5h-15a.5.5 0 0 0-.5.5V6h16ZM4 8v7.5a.5.5 0 0 0 .5.5h15a.5.5 0 0 0 .5-.5V8H4Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-delivery-medium" viewBox="0 0 24 24"><path d="M8.51 20.598a3.037 3.037 0 0 1-3.02 0A2.968 2.968 0 0 1 4.161 19L3.5 19A2.5 2.5 0 0 1 1 16.5v-11A2.5 2.5 0 0 1 3.5 3h10a2.5 2.5 0 0 1 2.45 2.004L16 5h2.527c.976 0 1.855.585 2.27 1.49l2.112 4.62a1 1 0 0 1 .091.416v4.856C23 17.814 21.889 19 20.484 19h-.523a1.01 1.01 0 0 1-.121-.007 2.96 2.96 0 0 1-1.33 1.605 3.037 3.037 0 0 1-3.02 0A2.968 2.968 0 0 1 14.161 19H9.838a2.968 2.968 0 0 1-1.327 1.597Zm-2.024-3.462a.955.955 0 0 0-.481.73L5.999 18l.001.022a.944.944 0 0 0 .388.777l.098.065c.316.181.712.181 1.028 0A.97.97 0 0 0 8 17.978a.95.95 0 0 0-.486-.842 1.037 1.037 0 0 0-1.028 0Zm10 0a.955.955 0 0 0-.481.73l-.005.156a.944.944 0 0 0 .388.777l.098.065c.316.181.712.181 1.028 0a.97.97 0 0 0 .486-.886.95.95 0 0 0-.486-.842 1.037 1.037 0 0 0-1.028 0ZM21 12h-5v3.17a3.038 3.038 0 0 1 2.51.232 2.993 2.993 0 0 1 1.277 1.45l.058.155.058-.005.581-.002c.27 0 .516-.263.516-.618V12Zm-7.5-7h-10a.5.5 0 0 0-.5.5v11a.5.5 0 0 0 .5.5h.662a2.964 2.964 0 0 1 1.155-1.491l.172-.107a3.037 3.037 0 0 1 3.022 0A2.987 2.987 0 0 1 9.843 17H13.5a.5.5 0 0 0 .5-.5v-11a.5.5 0 0 0-.5-.5Zm5.027 2H16v3h4.203l-1.224-2.677a.532.532 0 0 0-.375-.316L18.527 7Z"/></symbol><symbol id="icon-eds-i-download-medium" viewBox="0 0 24 24"><path d="M22 18.5a3.5 3.5 0 0 1-3.5 3.5h-13A3.5 3.5 0 0 1 2 18.5V18a1 1 0 0 1 2 0v.5A1.5 1.5 0 0 0 5.5 20h13a1.5 1.5 0 0 0 1.5-1.5V18a1 1 0 0 1 2 0v.5Zm-3.293-7.793-6 6-.063.059-.093.069-.081.048-.105.049-.104.034-.056.013-.118.017L12 17l-.076-.003-.122-.017-.113-.03-.085-.032-.063-.03-.098-.058-.06-.043-.05-.043-6.04-6.037a1 1 0 0 1 1.414-1.414l4.294 4.29L11 3a1 1 0 0 1 2 0l.001 10.585 4.292-4.292a1 1 0 0 1 1.32-.083l.094.083a1 1 0 0 1 0 1.414Z"/></symbol><symbol id="icon-eds-i-edit-medium" viewBox="0 0 24 24"><path d="M17.149 2a2.38 2.38 0 0 1 1.699.711l2.446 2.46a2.384 2.384 0 0 1 .005 3.38L10.01 19.906a1 1 0 0 1-.434.257l-6.3 1.8a1 1 0 0 1-1.237-1.237l1.8-6.3a1 1 0 0 1 .257-.434L15.443 2.718A2.385 2.385 0 0 1 17.15 2Zm-3.874 5.689-7.586 7.536-1.234 4.319 4.318-1.234 7.54-7.582-3.038-3.039ZM17.149 4a.395.395 0 0 0-.286.126L14.695 6.28l3.029 3.029 2.162-2.173a.384.384 0 0 0 .106-.197L20 6.864c0-.103-.04-.2-.119-.278l-2.457-2.47A.385.385 0 0 0 17.149 4Z"/></symbol><symbol id="icon-eds-i-education-medium" viewBox="0 0 24 24"><path fill-rule="evenodd" d="M12.41 2.088a1 1 0 0 0-.82 0l-10 4.5a1 1 0 0 0 0 1.824L3 9.047v7.124A3.001 3.001 0 0 0 4 22a3 3 0 0 0 1-5.83V9.948l1 .45V14.5a1 1 0 0 0 .087.408L7 14.5c-.913.408-.912.41-.912.41l.001.003.003.006.007.015a1.988 1.988 0 0 0 .083.16c.054.097.131.225.236.373.21.297.53.68.993 1.057C8.351 17.292 9.824 18 12 18c2.176 0 3.65-.707 4.589-1.476.463-.378.783-.76.993-1.057a4.162 4.162 0 0 0 .319-.533l.007-.015.003-.006v-.003h.002s0-.002-.913-.41l.913.408A1 1 0 0 0 18 14.5v-4.103l4.41-1.985a1 1 0 0 0 0-1.824l-10-4.5ZM16 11.297l-3.59 1.615a1 1 0 0 1-.82 0L8 11.297v2.94a3.388 3.388 0 0 0 .677.739C9.267 15.457 10.294 16 12 16s2.734-.543 3.323-1.024a3.388 3.388 0 0 0 .677-.739v-2.94ZM4.437 7.5 12 4.097 19.563 7.5 12 10.903 4.437 7.5ZM3 19a1 1 0 1 1 2 0 1 1 0 0 1-2 0Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-error-diamond-medium" viewBox="0 0 24 24"><path d="M12.002 1c.702 0 1.375.279 1.871.775l8.35 8.353a2.646 2.646 0 0 1 .001 3.744l-8.353 8.353a2.646 2.646 0 0 1-3.742 0l-8.353-8.353a2.646 2.646 0 0 1 0-3.744l8.353-8.353.156-.142c.424-.362.952-.58 1.507-.625l.21-.008Zm0 2a.646.646 0 0 0-.38.123l-.093.08-8.34 8.34a.646.646 0 0 0-.18.355L3 12c0 .171.068.336.19.457l8.353 8.354a.646.646 0 0 0 .914 0l8.354-8.354a.646.646 0 0 0-.001-.914l-8.351-8.354A.646.646 0 0 0 12.002 3ZM12 14.5a1.5 1.5 0 0 1 .144 2.993L12 17.5a1.5 1.5 0 0 1 0-3ZM12 6a1 1 0 0 1 1 1v5a1 1 0 0 1-2 0V7a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-error-filled-medium" viewBox="0 0 24 24"><path d="M12.002 1c.702 0 1.375.279 1.871.775l8.35 8.353a2.646 2.646 0 0 1 .001 3.744l-8.353 8.353a2.646 2.646 0 0 1-3.742 0l-8.353-8.353a2.646 2.646 0 0 1 0-3.744l8.353-8.353.156-.142c.424-.362.952-.58 1.507-.625l.21-.008ZM12 14.5a1.5 1.5 0 0 0 0 3l.144-.007A1.5 1.5 0 0 0 12 14.5ZM12 6a1 1 0 0 0-1 1v5a1 1 0 0 0 2 0V7a1 1 0 0 0-1-1Z"/></symbol><symbol id="icon-eds-i-external-link-medium" viewBox="0 0 24 24"><path d="M9 2a1 1 0 1 1 0 2H4.6c-.371 0-.6.209-.6.5v15c0 .291.229.5.6.5h14.8c.371 0 .6-.209.6-.5V15a1 1 0 0 1 2 0v4.5c0 1.438-1.162 2.5-2.6 2.5H4.6C3.162 22 2 20.938 2 19.5v-15C2 3.062 3.162 2 4.6 2H9Zm6 0h6l.075.003.126.017.111.03.111.044.098.052.096.067.09.08c.036.035.068.073.097.112l.071.11.054.114.035.105.03.148L22 3v6a1 1 0 0 1-2 0V5.414l-6.693 6.693a1 1 0 0 1-1.414-1.414L18.584 4H15a1 1 0 0 1-.993-.883L14 3a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-external-link-small" viewBox="0 0 16 16"><path d="M5 1a1 1 0 1 1 0 2l-2-.001V13L13 13v-2a1 1 0 0 1 2 0v2c0 1.15-.93 2-2.067 2H3.067C1.93 15 1 14.15 1 13V3c0-1.15.93-2 2.067-2H5Zm4 0h5l.075.003.126.017.111.03.111.044.098.052.096.067.09.08.044.047.073.093.051.083.054.113.035.105.03.148L15 2v5a1 1 0 0 1-2 0V4.414L9.107 8.307a1 1 0 0 1-1.414-1.414L11.584 3H9a1 1 0 0 1-.993-.883L8 2a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-file-download-medium" viewBox="0 0 24 24"><path d="M14.5 1a1 1 0 0 1 .707.293l5.5 5.5A1 1 0 0 1 21 7.5v12.962A2.542 2.542 0 0 1 18.455 23H5.545A2.542 2.542 0 0 1 3 20.462V3.538A2.542 2.542 0 0 1 5.545 1H14.5Zm-.415 2h-8.54A.542.542 0 0 0 5 3.538v16.924c0 .296.243.538.545.538h12.91a.542.542 0 0 0 .545-.538V7.915L14.085 3ZM12 7a1 1 0 0 1 1 1v6.585l2.293-2.292a1 1 0 0 1 1.32-.083l.094.083a1 1 0 0 1 0 1.414l-4 4a1.008 1.008 0 0 1-.112.097l-.11.071-.114.054-.105.035-.149.03L12 18l-.075-.003-.126-.017-.111-.03-.111-.044-.098-.052-.096-.067-.09-.08-4-4a1 1 0 0 1 1.414-1.414L11 14.585V8a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-file-report-medium" viewBox="0 0 24 24"><path d="M14.5 1a1 1 0 0 1 .707.293l5.5 5.5A1 1 0 0 1 21 7.5v12.962c0 .674-.269 1.32-.747 1.796a2.549 2.549 0 0 1-1.798.742H5.545c-.674 0-1.32-.267-1.798-.742A2.535 2.535 0 0 1 3 20.462V3.538A2.542 2.542 0 0 1 5.545 1H14.5Zm-.415 2h-8.54A.542.542 0 0 0 5 3.538v16.924c0 .142.057.278.158.379.102.102.242.159.387.159h12.91a.549.549 0 0 0 .387-.16.535.535 0 0 0 .158-.378V7.915L14.085 3ZM16 17a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h8Zm0-3a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h8Zm-4.793-6.207L13 9.585l1.793-1.792a1 1 0 0 1 1.32-.083l.094.083a1 1 0 0 1 0 1.414l-2.5 2.5a1 1 0 0 1-1.414 0L10.5 9.915l-1.793 1.792a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414l2.5-2.5a1 1 0 0 1 1.414 0Z"/></symbol><symbol id="icon-eds-i-file-text-medium" viewBox="0 0 24 24"><path d="M14.5 1a1 1 0 0 1 .707.293l5.5 5.5A1 1 0 0 1 21 7.5v12.962A2.542 2.542 0 0 1 18.455 23H5.545A2.542 2.542 0 0 1 3 20.462V3.538A2.542 2.542 0 0 1 5.545 1H14.5Zm-.415 2h-8.54A.542.542 0 0 0 5 3.538v16.924c0 .296.243.538.545.538h12.91a.542.542 0 0 0 .545-.538V7.915L14.085 3ZM16 15a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h8Zm0-4a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h8Zm-5-4a1 1 0 0 1 0 2H8a1 1 0 1 1 0-2h3Z"/></symbol><symbol id="icon-eds-i-file-upload-medium" viewBox="0 0 24 24"><path d="M14.5 1a1 1 0 0 1 .707.293l5.5 5.5A1 1 0 0 1 21 7.5v12.962A2.542 2.542 0 0 1 18.455 23H5.545A2.542 2.542 0 0 1 3 20.462V3.538A2.542 2.542 0 0 1 5.545 1H14.5Zm-.415 2h-8.54A.542.542 0 0 0 5 3.538v16.924c0 .296.243.538.545.538h12.91a.542.542 0 0 0 .545-.538V7.915L14.085 3Zm-2.233 4.011.058-.007L12 7l.075.003.126.017.111.03.111.044.098.052.104.074.082.073 4 4a1 1 0 0 1 0 1.414l-.094.083a1 1 0 0 1-1.32-.083L13 10.415V17a1 1 0 0 1-2 0v-6.585l-2.293 2.292a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414l4-4 .112-.097.11-.071.114-.054.105-.035.118-.025Z"/></symbol><symbol id="icon-eds-i-filter-medium" viewBox="0 0 24 24"><path d="M21 2a1 1 0 0 1 .82 1.573L15 13.314V18a1 1 0 0 1-.31.724l-.09.076-4 3A1 1 0 0 1 9 21v-7.684L2.18 3.573a1 1 0 0 1 .707-1.567L3 2h18Zm-1.921 2H4.92l5.9 8.427a1 1 0 0 1 .172.45L11 13v6l2-1.5V13a1 1 0 0 1 .117-.469l.064-.104L19.079 4Z"/></symbol><symbol id="icon-eds-i-funding-medium" viewBox="0 0 24 24"><path fill-rule="evenodd" d="M23 8A7 7 0 1 0 9 8a7 7 0 0 0 14 0ZM9.006 12.225A4.07 4.07 0 0 0 6.12 11.02H2a.979.979 0 1 0 0 1.958h4.12c.558 0 1.094.222 1.489.617l2.207 2.288c.27.27.27.687.012.944a.656.656 0 0 1-.928 0L7.744 15.67a.98.98 0 0 0-1.386 1.384l1.157 1.158c.535.536 1.244.791 1.946.765l.041.002h6.922c.874 0 1.597.748 1.597 1.688 0 .203-.146.354-.309.354H7.755c-.487 0-.96-.178-1.339-.504L2.64 17.259a.979.979 0 0 0-1.28 1.482L5.137 22c.733.631 1.66.979 2.618.979h9.957c1.26 0 2.267-1.043 2.267-2.312 0-2.006-1.584-3.646-3.555-3.646h-4.529a2.617 2.617 0 0 0-.681-2.509l-2.208-2.287ZM16 3a5 5 0 1 0 0 10 5 5 0 0 0 0-10Zm.979 3.5a.979.979 0 1 0-1.958 0v3a.979.979 0 1 0 1.958 0v-3Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-hashtag-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 2a9 9 0 1 0 0 18 9 9 0 0 0 0-18ZM9.52 18.189a1 1 0 1 1-1.964-.378l.437-2.274H6a1 1 0 1 1 0-2h2.378l.592-3.076H6a1 1 0 0 1 0-2h3.354l.51-2.65a1 1 0 1 1 1.964.378l-.437 2.272h3.04l.51-2.65a1 1 0 1 1 1.964.378l-.438 2.272H18a1 1 0 0 1 0 2h-1.917l-.592 3.076H18a1 1 0 0 1 0 2h-2.893l-.51 2.652a1 1 0 1 1-1.964-.378l.437-2.274h-3.04l-.51 2.652Zm.895-4.652h3.04l.591-3.076h-3.04l-.591 3.076Z"/></symbol><symbol id="icon-eds-i-home-medium" viewBox="0 0 24 24"><path d="M5 22a1 1 0 0 1-1-1v-8.586l-1.293 1.293a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414l10-10a1 1 0 0 1 1.414 0l10 10a1 1 0 0 1-1.414 1.414L20 12.415V21a1 1 0 0 1-1 1H5Zm7-17.585-6 5.999V20h5v-4a1 1 0 0 1 2 0v4h5v-9.585l-6-6Z"/></symbol><symbol id="icon-eds-i-image-medium" viewBox="0 0 24 24"><path d="M19.615 2A2.385 2.385 0 0 1 22 4.385v15.23A2.385 2.385 0 0 1 19.615 22H4.385A2.385 2.385 0 0 1 2 19.615V4.385A2.385 2.385 0 0 1 4.385 2h15.23Zm0 2H4.385A.385.385 0 0 0 4 4.385v15.23c0 .213.172.385.385.385h1.244l10.228-8.76a1 1 0 0 1 1.254-.037L20 13.392V4.385A.385.385 0 0 0 19.615 4Zm-3.07 9.283L8.703 20h10.912a.385.385 0 0 0 .385-.385v-3.713l-3.455-2.619ZM9.5 6a3.5 3.5 0 1 1 0 7 3.5 3.5 0 0 1 0-7Zm0 2a1.5 1.5 0 1 0 0 3 1.5 1.5 0 0 0 0-3Z"/></symbol><symbol id="icon-eds-i-impact-factor-medium" viewBox="0 0 24 24"><path d="M16.49 2.672c.74.694.986 1.765.632 2.712l-.04.1-1.549 3.54h1.477a2.496 2.496 0 0 1 2.485 2.34l.005.163c0 .618-.23 1.21-.642 1.675l-7.147 7.961a2.48 2.48 0 0 1-3.554.165 2.512 2.512 0 0 1-.633-2.712l.042-.103L9.108 15H7.46c-1.393 0-2.379-1.11-2.455-2.369L5 12.473c0-.593.142-1.145.628-1.692l7.307-7.944a2.48 2.48 0 0 1 3.555-.165ZM14.43 4.164l-7.33 7.97c-.083.093-.101.214-.101.34 0 .277.19.526.46.526h4.163l.097-.009c.015 0 .03.003.046.009.181.078.264.32.186.5l-2.554 5.817a.512.512 0 0 0 .127.552.48.48 0 0 0 .69-.033l7.155-7.97a.513.513 0 0 0 .13-.34.497.497 0 0 0-.49-.502h-3.988a.355.355 0 0 1-.328-.497l2.555-5.844a.512.512 0 0 0-.127-.552.48.48 0 0 0-.69.033Z"/></symbol><symbol id="icon-eds-i-info-circle-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 2a9 9 0 1 0 0 18 9 9 0 0 0 0-18Zm0 7a1 1 0 0 1 1 1v5h1.5a1 1 0 0 1 0 2h-5a1 1 0 0 1 0-2H11v-4h-.5a1 1 0 0 1-.993-.883L9.5 11a1 1 0 0 1 1-1H12Zm0-4.5a1.5 1.5 0 0 1 .144 2.993L12 8.5a1.5 1.5 0 0 1 0-3Z"/></symbol><symbol id="icon-eds-i-info-filled-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 9h-1.5a1 1 0 0 0-1 1l.007.117A1 1 0 0 0 10.5 12h.5v4H9.5a1 1 0 0 0 0 2h5a1 1 0 0 0 0-2H13v-5a1 1 0 0 0-1-1Zm0-4.5a1.5 1.5 0 0 0 0 3l.144-.007A1.5 1.5 0 0 0 12 5.5Z"/></symbol><symbol id="icon-eds-i-journal-medium" viewBox="0 0 24 24"><path d="M18.5 1A2.5 2.5 0 0 1 21 3.5v14a2.5 2.5 0 0 1-2.5 2.5h-13a.5.5 0 1 0 0 1H20a1 1 0 0 1 0 2H5.5A2.5 2.5 0 0 1 3 20.5v-17A2.5 2.5 0 0 1 5.5 1h13ZM7 3H5.5a.5.5 0 0 0-.5.5v14.549l.016-.002c.104-.02.211-.035.32-.042L5.5 18H7V3Zm11.5 0H9v15h9.5a.5.5 0 0 0 .5-.5v-14a.5.5 0 0 0-.5-.5ZM16 5a1 1 0 0 1 1 1v4a1 1 0 0 1-1 1h-5a1 1 0 0 1-1-1V6a1 1 0 0 1 1-1h5Zm-1 2h-3v2h3V7Z"/></symbol><symbol id="icon-eds-i-mail-medium" viewBox="0 0 24 24"><path d="M20.462 3C21.875 3 23 4.184 23 5.619v12.762C23 19.816 21.875 21 20.462 21H3.538C2.125 21 1 19.816 1 18.381V5.619C1 4.184 2.125 3 3.538 3h16.924ZM21 8.158l-7.378 6.258a2.549 2.549 0 0 1-3.253-.008L3 8.16v10.222c0 .353.253.619.538.619h16.924c.285 0 .538-.266.538-.619V8.158ZM20.462 5H3.538c-.264 0-.5.228-.534.542l8.65 7.334c.2.165.492.165.684.007l8.656-7.342-.001-.025c-.044-.3-.274-.516-.531-.516Z"/></symbol><symbol id="icon-eds-i-mail-send-medium" viewBox="0 0 24 24"><path d="M20.444 5a2.562 2.562 0 0 1 2.548 2.37l.007.078.001.123v7.858A2.564 2.564 0 0 1 20.444 18H9.556A2.564 2.564 0 0 1 7 15.429l.001-7.977.007-.082A2.561 2.561 0 0 1 9.556 5h10.888ZM21 9.331l-5.46 3.51a1 1 0 0 1-1.08 0L9 9.332v6.097c0 .317.251.571.556.571h10.888a.564.564 0 0 0 .556-.571V9.33ZM20.444 7H9.556a.543.543 0 0 0-.32.105l5.763 3.706 5.766-3.706a.543.543 0 0 0-.32-.105ZM4.308 5a1 1 0 1 1 0 2H2a1 1 0 1 1 0-2h2.308Zm0 5.5a1 1 0 0 1 0 2H2a1 1 0 0 1 0-2h2.308Zm0 5.5a1 1 0 0 1 0 2H2a1 1 0 0 1 0-2h2.308Z"/></symbol><symbol id="icon-eds-i-mentions-medium" viewBox="0 0 24 24"><path d="m9.452 1.293 5.92 5.92 2.92-2.92a1 1 0 0 1 1.415 1.414l-2.92 2.92 5.92 5.92a1 1 0 0 1 0 1.415 10.371 10.371 0 0 1-10.378 2.584l.652 3.258A1 1 0 0 1 12 23H2a1 1 0 0 1-.874-1.486l4.789-8.62C4.194 9.074 4.9 4.43 8.038 1.292a1 1 0 0 1 1.414 0Zm-2.355 13.59L3.699 21h7.081l-.689-3.442a10.392 10.392 0 0 1-2.775-2.396l-.22-.28Zm1.69-11.427-.07.09a8.374 8.374 0 0 0 11.737 11.737l.089-.071L8.787 3.456Z"/></symbol><symbol id="icon-eds-i-menu-medium" viewBox="0 0 24 24"><path d="M21 4a1 1 0 0 1 0 2H3a1 1 0 1 1 0-2h18Zm-4 7a1 1 0 0 1 0 2H3a1 1 0 0 1 0-2h14Zm4 7a1 1 0 0 1 0 2H3a1 1 0 0 1 0-2h18Z"/></symbol><symbol id="icon-eds-i-metrics-medium" viewBox="0 0 24 24"><path d="M3 22a1 1 0 0 1-1-1V3a1 1 0 0 1 1-1h6a1 1 0 0 1 1 1v7h4V8a1 1 0 0 1 1-1h6a1 1 0 0 1 1 1v13a1 1 0 0 1-.883.993L21 22H3Zm17-2V9h-4v11h4Zm-6-8h-4v8h4v-8ZM8 4H4v16h4V4Z"/></symbol><symbol id="icon-eds-i-news-medium" viewBox="0 0 24 24"><path d="M17.384 3c.975 0 1.77.787 1.77 1.762v13.333c0 .462.354.846.815.899l.107.006.109-.006a.915.915 0 0 0 .809-.794l.006-.105V8.19a1 1 0 0 1 2 0v9.905A2.914 2.914 0 0 1 20.077 21H3.538a2.547 2.547 0 0 1-1.644-.601l-.147-.135A2.516 2.516 0 0 1 1 18.476V4.762C1 3.787 1.794 3 2.77 3h14.614Zm-.231 2H3v13.476c0 .11.035.216.1.304l.054.063c.101.1.24.157.384.157l13.761-.001-.026-.078a2.88 2.88 0 0 1-.115-.655l-.004-.17L17.153 5ZM14 15.021a.979.979 0 1 1 0 1.958H6a.979.979 0 1 1 0-1.958h8Zm0-8c.54 0 .979.438.979.979v4c0 .54-.438.979-.979.979H6A.979.979 0 0 1 5.021 12V8c0-.54.438-.979.979-.979h8Zm-.98 1.958H6.979v2.041h6.041V8.979Z"/></symbol><symbol id="icon-eds-i-newsletter-medium" viewBox="0 0 24 24"><path d="M21 10a1 1 0 0 1 1 1v9.5a2.5 2.5 0 0 1-2.5 2.5h-15A2.5 2.5 0 0 1 2 20.5V11a1 1 0 0 1 2 0v.439l8 4.888 8-4.889V11a1 1 0 0 1 1-1Zm-1 3.783-7.479 4.57a1 1 0 0 1-1.042 0l-7.48-4.57V20.5a.5.5 0 0 0 .501.5h15a.5.5 0 0 0 .5-.5v-6.717ZM15 9a1 1 0 0 1 0 2H9a1 1 0 0 1 0-2h6Zm2.5-8A2.5 2.5 0 0 1 20 3.5V9a1 1 0 0 1-2 0V3.5a.5.5 0 0 0-.5-.5h-11a.5.5 0 0 0-.5.5V9a1 1 0 1 1-2 0V3.5A2.5 2.5 0 0 1 6.5 1h11ZM15 5a1 1 0 0 1 0 2H9a1 1 0 1 1 0-2h6Z"/></symbol><symbol id="icon-eds-i-notifcation-medium" viewBox="0 0 24 24"><path d="M14 20a1 1 0 0 1 0 2h-4a1 1 0 0 1 0-2h4ZM3 18l-.133-.007c-1.156-.124-1.156-1.862 0-1.986l.3-.012C4.32 15.923 5 15.107 5 14V9.5C5 5.368 8.014 2 12 2s7 3.368 7 7.5V14c0 1.107.68 1.923 1.832 1.995l.301.012c1.156.124 1.156 1.862 0 1.986L21 18H3Zm9-14C9.17 4 7 6.426 7 9.5V14c0 .671-.146 1.303-.416 1.858L6.51 16h10.979l-.073-.142a4.192 4.192 0 0 1-.412-1.658L17 14V9.5C17 6.426 14.83 4 12 4Z"/></symbol><symbol id="icon-eds-i-publish-medium" viewBox="0 0 24 24"><g><path d="M16.296 1.291A1 1 0 0 0 15.591 1H5.545A2.542 2.542 0 0 0 3 3.538V13a1 1 0 1 0 2 0V3.538l.007-.087A.543.543 0 0 1 5.545 3h9.633L20 7.8v12.662a.534.534 0 0 1-.158.379.548.548 0 0 1-.387.159H11a1 1 0 1 0 0 2h8.455c.674 0 1.32-.267 1.798-.742A2.534 2.534 0 0 0 22 20.462V7.385a1 1 0 0 0-.294-.709l-5.41-5.385Z"/><path d="M10.762 16.647a1 1 0 0 0-1.525-1.294l-4.472 5.271-2.153-1.665a1 1 0 1 0-1.224 1.582l2.91 2.25a1 1 0 0 0 1.374-.144l5.09-6ZM16 10a1 1 0 1 1 0 2H8a1 1 0 1 1 0-2h8ZM12 7a1 1 0 0 0-1-1H8a1 1 0 1 0 0 2h3a1 1 0 0 0 1-1Z"/></g></symbol><symbol id="icon-eds-i-refresh-medium" viewBox="0 0 24 24"><g><path d="M7.831 5.636H6.032A8.76 8.76 0 0 1 9 3.631 8.549 8.549 0 0 1 12.232 3c.603 0 1.192.063 1.76.182C17.979 4.017 21 7.632 21 12a1 1 0 1 0 2 0c0-5.296-3.674-9.746-8.591-10.776A10.61 10.61 0 0 0 5 3.851V2.805a1 1 0 0 0-.987-1H4a1 1 0 0 0-1 1v3.831a1 1 0 0 0 1 1h3.831a1 1 0 0 0 .013-2h-.013ZM17.968 18.364c-1.59 1.632-3.784 2.636-6.2 2.636C6.948 21 3 16.993 3 12a1 1 0 1 0-2 0c0 6.053 4.799 11 10.768 11 2.788 0 5.324-1.082 7.232-2.85v1.045a1 1 0 1 0 2 0v-3.831a1 1 0 0 0-1-1h-3.831a1 1 0 0 0 0 2h1.799Z"/></g></symbol><symbol id="icon-eds-i-search-medium" viewBox="0 0 24 24"><path d="M11 1c5.523 0 10 4.477 10 10 0 2.4-.846 4.604-2.256 6.328l3.963 3.965a1 1 0 0 1-1.414 1.414l-3.965-3.963A9.959 9.959 0 0 1 11 21C5.477 21 1 16.523 1 11S5.477 1 11 1Zm0 2a8 8 0 1 0 0 16 8 8 0 0 0 0-16Z"/></symbol><symbol id="icon-eds-i-settings-medium" viewBox="0 0 24 24"><path d="M11.382 1h1.24a2.508 2.508 0 0 1 2.334 1.63l.523 1.378 1.59.933 1.444-.224c.954-.132 1.89.3 2.422 1.101l.095.155.598 1.066a2.56 2.56 0 0 1-.195 2.848l-.894 1.161v1.896l.92 1.163c.6.768.707 1.812.295 2.674l-.09.17-.606 1.08a2.504 2.504 0 0 1-2.531 1.25l-1.428-.223-1.589.932-.523 1.378a2.512 2.512 0 0 1-2.155 1.625L12.65 23h-1.27a2.508 2.508 0 0 1-2.334-1.63l-.524-1.379-1.59-.933-1.443.225c-.954.132-1.89-.3-2.422-1.101l-.095-.155-.598-1.066a2.56 2.56 0 0 1 .195-2.847l.891-1.161v-1.898l-.919-1.162a2.562 2.562 0 0 1-.295-2.674l.09-.17.606-1.08a2.504 2.504 0 0 1 2.531-1.25l1.43.223 1.618-.938.524-1.375.07-.167A2.507 2.507 0 0 1 11.382 1Zm.003 2a.509.509 0 0 0-.47.338l-.65 1.71a1 1 0 0 1-.434.51L7.6 6.85a1 1 0 0 1-.655.123l-1.762-.275a.497.497 0 0 0-.498.252l-.61 1.088a.562.562 0 0 0 .04.619l1.13 1.43a1 1 0 0 1 .216.62v2.585a1 1 0 0 1-.207.61L4.15 15.339a.568.568 0 0 0-.036.634l.601 1.072a.494.494 0 0 0 .484.26l1.78-.278a1 1 0 0 1 .66.126l2.2 1.292a1 1 0 0 1 .43.507l.648 1.71a.508.508 0 0 0 .467.338h1.263a.51.51 0 0 0 .47-.34l.65-1.708a1 1 0 0 1 .428-.507l2.201-1.292a1 1 0 0 1 .66-.126l1.763.275a.497.497 0 0 0 .498-.252l.61-1.088a.562.562 0 0 0-.04-.619l-1.13-1.43a1 1 0 0 1-.216-.62v-2.585a1 1 0 0 1 .207-.61l1.105-1.437a.568.568 0 0 0 .037-.634l-.601-1.072a.494.494 0 0 0-.484-.26l-1.78.278a1 1 0 0 1-.66-.126l-2.2-1.292a1 1 0 0 1-.43-.507l-.649-1.71A.508.508 0 0 0 12.62 3h-1.234ZM12 8a4 4 0 1 1 0 8 4 4 0 0 1 0-8Zm0 2a2 2 0 1 0 0 4 2 2 0 0 0 0-4Z"/></symbol><symbol id="icon-eds-i-shipping-medium" viewBox="0 0 24 24"><path d="M16.515 2c1.406 0 2.706.728 3.352 1.902l2.02 3.635.02.042.036.089.031.105.012.058.01.073.004.075v11.577c0 .64-.244 1.255-.683 1.713a2.356 2.356 0 0 1-1.701.731H4.386a2.356 2.356 0 0 1-1.702-.731 2.476 2.476 0 0 1-.683-1.713V7.948c.01-.217.083-.43.22-.6L4.2 3.905C4.833 2.755 6.089 2.032 7.486 2h9.029ZM20 9H4v10.556a.49.49 0 0 0 .075.26l.053.07a.356.356 0 0 0 .257.114h15.23c.094 0 .186-.04.258-.115a.477.477 0 0 0 .127-.33V9Zm-2 7.5a1 1 0 0 1 0 2h-4a1 1 0 0 1 0-2h4ZM16.514 4H13v3h6.3l-1.183-2.13c-.288-.522-.908-.87-1.603-.87ZM11 3.999H7.51c-.679.017-1.277.36-1.566.887L4.728 7H11V3.999Z"/></symbol><symbol id="icon-eds-i-step-guide-medium" viewBox="0 0 24 24"><path d="M11.394 9.447a1 1 0 1 0-1.788-.894l-.88 1.759-.019-.02a1 1 0 1 0-1.414 1.415l1 1a1 1 0 0 0 1.601-.26l1.5-3ZM12 11a1 1 0 0 1 1-1h3a1 1 0 1 1 0 2h-3a1 1 0 0 1-1-1ZM12 17a1 1 0 0 1 1-1h3a1 1 0 1 1 0 2h-3a1 1 0 0 1-1-1ZM10.947 14.105a1 1 0 0 1 .447 1.342l-1.5 3a1 1 0 0 1-1.601.26l-1-1a1 1 0 1 1 1.414-1.414l.02.019.879-1.76a1 1 0 0 1 1.341-.447Z"/><path d="M5.545 1A2.542 2.542 0 0 0 3 3.538v16.924A2.542 2.542 0 0 0 5.545 23h12.91A2.542 2.542 0 0 0 21 20.462V7.5a1 1 0 0 0-.293-.707l-5.5-5.5A1 1 0 0 0 14.5 1H5.545ZM5 3.538C5 3.245 5.24 3 5.545 3h8.54L19 7.914v12.547c0 .294-.24.539-.546.539H5.545A.542.542 0 0 1 5 20.462V3.538Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-submission-medium" viewBox="0 0 24 24"><g><path d="M5 3.538C5 3.245 5.24 3 5.545 3h9.633L20 7.8v12.662a.535.535 0 0 1-.158.379.549.549 0 0 1-.387.159H6a1 1 0 0 1-1-1v-2.5a1 1 0 1 0-2 0V20a3 3 0 0 0 3 3h13.455c.673 0 1.32-.266 1.798-.742A2.535 2.535 0 0 0 22 20.462V7.385a1 1 0 0 0-.294-.709l-5.41-5.385A1 1 0 0 0 15.591 1H5.545A2.542 2.542 0 0 0 3 3.538V7a1 1 0 0 0 2 0V3.538Z"/><path d="m13.707 13.707-4 4a1 1 0 0 1-1.414 0l-.083-.094a1 1 0 0 1 .083-1.32L10.585 14 2 14a1 1 0 1 1 0-2l8.583.001-2.29-2.294a1 1 0 0 1 1.414-1.414l4.037 4.04.043.05.043.06.059.098.03.063.031.085.03.113.017.122L14 13l-.004.087-.017.118-.013.056-.034.104-.049.105-.048.081-.07.093-.058.063Z"/></g></symbol><symbol id="icon-eds-i-table-1-medium" viewBox="0 0 24 24"><path d="M4.385 22a2.56 2.56 0 0 1-1.14-.279C2.485 21.341 2 20.614 2 19.615V4.385c0-.315.067-.716.279-1.14C2.659 2.485 3.386 2 4.385 2h15.23c.315 0 .716.067 1.14.279.76.38 1.245 1.107 1.245 2.106v15.23c0 .315-.067.716-.279 1.14-.38.76-1.107 1.245-2.106 1.245H4.385ZM4 19.615c0 .213.034.265.14.317a.71.71 0 0 0 .245.068H8v-4H4v3.615ZM20 16H10v4h9.615c.213 0 .265-.034.317-.14a.71.71 0 0 0 .068-.245V16Zm0-2v-4H10v4h10ZM4 14h4v-4H4v4ZM19.615 4H10v4h10V4.385c0-.213-.034-.265-.14-.317A.71.71 0 0 0 19.615 4ZM8 4H4.385l-.082.002c-.146.01-.19.047-.235.138A.71.71 0 0 0 4 4.385V8h4V4Z"/></symbol><symbol id="icon-eds-i-table-2-medium" viewBox="0 0 24 24"><path d="M4.384 22A2.384 2.384 0 0 1 2 19.616V4.384A2.384 2.384 0 0 1 4.384 2h15.232A2.384 2.384 0 0 1 22 4.384v15.232A2.384 2.384 0 0 1 19.616 22H4.384ZM10 15H4v4.616c0 .212.172.384.384.384H10v-5Zm5 0h-3v5h3v-5Zm5 0h-3v5h2.616a.384.384 0 0 0 .384-.384V15ZM10 9H4v4h6V9Zm5 0h-3v4h3V9Zm5 0h-3v4h3V9Zm-.384-5H4.384A.384.384 0 0 0 4 4.384V7h16V4.384A.384.384 0 0 0 19.616 4Z"/></symbol><symbol id="icon-eds-i-tag-medium" viewBox="0 0 24 24"><path d="m12.621 1.998.127.004L20.496 2a1.5 1.5 0 0 1 1.497 1.355L22 3.5l-.005 7.669c.038.456-.133.905-.447 1.206l-9.02 9.018a2.075 2.075 0 0 1-2.932 0l-6.99-6.99a2.075 2.075 0 0 1 .001-2.933L11.61 2.47c.246-.258.573-.418.881-.46l.131-.011Zm.286 2-8.885 8.886a.075.075 0 0 0 0 .106l6.987 6.988c.03.03.077.03.106 0l8.883-8.883L19.999 4l-7.092-.002ZM16 6.5a1.5 1.5 0 0 1 .144 2.993L16 9.5a1.5 1.5 0 0 1 0-3Z"/></symbol><symbol id="icon-eds-i-trash-medium" viewBox="0 0 24 24"><path d="M12 1c2.717 0 4.913 2.232 4.997 5H21a1 1 0 0 1 0 2h-1v12.5c0 1.389-1.152 2.5-2.556 2.5H6.556C5.152 23 4 21.889 4 20.5V8H3a1 1 0 1 1 0-2h4.003l.001-.051C7.114 3.205 9.3 1 12 1Zm6 7H6v12.5c0 .238.19.448.454.492l.102.008h10.888c.315 0 .556-.232.556-.5V8Zm-4 3a1 1 0 0 1 1 1v6.005a1 1 0 0 1-2 0V12a1 1 0 0 1 1-1Zm-4 0a1 1 0 0 1 1 1v6a1 1 0 0 1-2 0v-6a1 1 0 0 1 1-1Zm2-8c-1.595 0-2.914 1.32-2.996 3h5.991v-.02C14.903 4.31 13.589 3 12 3Z"/></symbol><symbol id="icon-eds-i-user-account-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 16c-1.806 0-3.52.994-4.664 2.698A8.947 8.947 0 0 0 12 21a8.958 8.958 0 0 0 4.664-1.301C15.52 17.994 13.806 17 12 17Zm0-14a9 9 0 0 0-6.25 15.476C7.253 16.304 9.54 15 12 15s4.747 1.304 6.25 3.475A9 9 0 0 0 12 3Zm0 3a4 4 0 1 1 0 8 4 4 0 0 1 0-8Zm0 2a2 2 0 1 0 0 4 2 2 0 0 0 0-4Z"/></symbol><symbol id="icon-eds-i-user-add-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm9 10a1 1 0 0 1 1 1v3h3a1 1 0 0 1 0 2h-3v3a1 1 0 0 1-2 0v-3h-3a1 1 0 0 1 0-2h3v-3a1 1 0 0 1 1-1Zm-5.545-.15a1 1 0 1 1-.91 1.78 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 11.5 20a1 1 0 0 1 .993.883L12.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378Z"/></symbol><symbol id="icon-eds-i-user-assign-medium" viewBox="0 0 24 24"><path d="M16.226 13.298a1 1 0 0 1 1.414-.01l.084.093a1 1 0 0 1-.073 1.32L15.39 17H22a1 1 0 0 1 0 2h-6.611l2.262 2.298a1 1 0 0 1-1.425 1.404l-3.939-4a1 1 0 0 1 0-1.404l3.94-4Zm-3.771-.449a1 1 0 1 1-.91 1.781 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 10.5 20a1 1 0 0 1 .993.883L11.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378ZM9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Z"/></symbol><symbol id="icon-eds-i-user-block-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm9 10a5 5 0 1 1 0 10 5 5 0 0 1 0-10Zm-5.545-.15a1 1 0 1 1-.91 1.78 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 11.5 20a1 1 0 0 1 .993.883L12.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378ZM15 18a3 3 0 0 0 4.294 2.707l-4.001-4c-.188.391-.293.83-.293 1.293Zm3-3c-.463 0-.902.105-1.294.293l4.001 4A3 3 0 0 0 18 15Z"/></symbol><symbol id="icon-eds-i-user-check-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm13.647 12.237a1 1 0 0 1 .116 1.41l-5.091 6a1 1 0 0 1-1.375.144l-2.909-2.25a1 1 0 1 1 1.224-1.582l2.153 1.665 4.472-5.271a1 1 0 0 1 1.41-.116Zm-8.139-.977c.22.214.428.44.622.678a1 1 0 1 1-1.548 1.266 6.025 6.025 0 0 0-1.795-1.49.86.86 0 0 1-.163-.048l-.079-.036a5.721 5.721 0 0 0-2.62-.63l-.194.006c-2.76.134-5.022 2.177-5.592 4.864l-.035.175-.035.213c-.03.201-.05.405-.06.61L3.003 20 10 20a1 1 0 0 1 .993.883L11 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876l.005-.223.02-.356.02-.222.03-.248.022-.15c.02-.133.044-.265.071-.397.44-2.178 1.725-4.105 3.595-5.301a7.75 7.75 0 0 1 3.755-1.215l.12-.004a7.908 7.908 0 0 1 5.87 2.252Z"/></symbol><symbol id="icon-eds-i-user-delete-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6ZM4.763 13.227a7.713 7.713 0 0 1 7.692-.378 1 1 0 1 1-.91 1.781 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20H11.5a1 1 0 0 1 .993.883L12.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897Zm11.421 1.543 2.554 2.553 2.555-2.553a1 1 0 0 1 1.414 1.414l-2.554 2.554 2.554 2.555a1 1 0 0 1-1.414 1.414l-2.555-2.554-2.554 2.554a1 1 0 0 1-1.414-1.414l2.553-2.555-2.553-2.554a1 1 0 0 1 1.414-1.414Z"/></symbol><symbol id="icon-eds-i-user-edit-medium" viewBox="0 0 24 24"><path d="m19.876 10.77 2.831 2.83a1 1 0 0 1 0 1.415l-7.246 7.246a1 1 0 0 1-.572.284l-3.277.446a1 1 0 0 1-1.125-1.13l.461-3.277a1 1 0 0 1 .283-.567l7.23-7.246a1 1 0 0 1 1.415-.001Zm-7.421 2.08a1 1 0 1 1-.91 1.78 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 7.5 20a1 1 0 0 1 .993.883L8.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378Zm6.715.042-6.29 6.3-.23 1.639 1.633-.222 6.302-6.302-1.415-1.415ZM9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Z"/></symbol><symbol id="icon-eds-i-user-linked-medium" viewBox="0 0 24 24"><path d="M15.65 6c.31 0 .706.066 1.122.274C17.522 6.65 18 7.366 18 8.35v12.3c0 .31-.066.706-.274 1.122-.375.75-1.092 1.228-2.076 1.228H3.35a2.52 2.52 0 0 1-1.122-.274C1.478 22.35 1 21.634 1 20.65V8.35c0-.31.066-.706.274-1.122C1.65 6.478 2.366 6 3.35 6h12.3Zm0 2-12.376.002c-.134.007-.17.04-.21.12A.672.672 0 0 0 3 8.35v12.3c0 .198.028.24.122.287.09.044.2.063.228.063h.887c.788-2.269 2.814-3.5 5.263-3.5 2.45 0 4.475 1.231 5.263 3.5h.887c.198 0 .24-.028.287-.122.044-.09.063-.2.063-.228V8.35c0-.198-.028-.24-.122-.287A.672.672 0 0 0 15.65 8ZM9.5 19.5c-1.36 0-2.447.51-3.06 1.5h6.12c-.613-.99-1.7-1.5-3.06-1.5ZM20.65 1A2.35 2.35 0 0 1 23 3.348V15.65A2.35 2.35 0 0 1 20.65 18H20a1 1 0 0 1 0-2h.65a.35.35 0 0 0 .35-.35V3.348A.35.35 0 0 0 20.65 3H8.35a.35.35 0 0 0-.35.348V4a1 1 0 1 1-2 0v-.652A2.35 2.35 0 0 1 8.35 1h12.3ZM9.5 10a3.5 3.5 0 1 1 0 7 3.5 3.5 0 0 1 0-7Zm0 2a1.5 1.5 0 1 0 0 3 1.5 1.5 0 0 0 0-3Z"/></symbol><symbol id="icon-eds-i-user-multiple-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm6 0a5 5 0 0 1 0 10 1 1 0 0 1-.117-1.993L15 9a3 3 0 0 0 0-6 1 1 0 0 1 0-2ZM9 3a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm8.857 9.545a7.99 7.99 0 0 1 2.651 1.715A8.31 8.31 0 0 1 23 20.134V21a1 1 0 0 1-1 1h-3a1 1 0 0 1 0-2h1.995l-.005-.153a6.307 6.307 0 0 0-1.673-3.945l-.204-.209a5.99 5.99 0 0 0-1.988-1.287 1 1 0 1 1 .732-1.861Zm-3.349 1.715A8.31 8.31 0 0 1 17 20.134V21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.877c.044-4.343 3.387-7.908 7.638-8.115a7.908 7.908 0 0 1 5.87 2.252ZM9.016 14l-.285.006c-3.104.15-5.58 2.718-5.725 5.9L3.004 20h11.991l-.005-.153a6.307 6.307 0 0 0-1.673-3.945l-.204-.209A5.924 5.924 0 0 0 9.3 14.008L9.016 14Z"/></symbol><symbol id="icon-eds-i-user-notify-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm10 18v1a1 1 0 0 1-2 0v-1h-3a1 1 0 0 1 0-2v-2.818C14 13.885 15.777 12 18 12s4 1.885 4 4.182V19a1 1 0 0 1 0 2h-3Zm-6.545-8.15a1 1 0 1 1-.91 1.78 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 11.5 20a1 1 0 0 1 .993.883L12.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378ZM18 14c-1.091 0-2 .964-2 2.182V19h4v-2.818c0-1.165-.832-2.098-1.859-2.177L18 14Z"/></symbol><symbol id="icon-eds-i-user-remove-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm3.455 9.85a1 1 0 1 1-.91 1.78 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 11.5 20a1 1 0 0 1 .993.883L12.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378ZM22 17a1 1 0 0 1 0 2h-8a1 1 0 0 1 0-2h8Z"/></symbol><symbol id="icon-eds-i-user-single-medium" viewBox="0 0 24 24"><path d="M12 1a5 5 0 1 1 0 10 5 5 0 0 1 0-10Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm-.406 9.008a8.965 8.965 0 0 1 6.596 2.494A9.161 9.161 0 0 1 21 21.025V22a1 1 0 0 1-1 1H4a1 1 0 0 1-1-1v-.985c.05-4.825 3.815-8.777 8.594-9.007Zm.39 1.992-.299.006c-3.63.175-6.518 3.127-6.678 6.775L5 21h13.998l-.009-.268a7.157 7.157 0 0 0-1.97-4.573l-.214-.213A6.967 6.967 0 0 0 11.984 14Z"/></symbol><symbol id="icon-eds-i-warning-circle-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 2a9 9 0 1 0 0 18 9 9 0 0 0 0-18Zm0 11.5a1.5 1.5 0 0 1 .144 2.993L12 17.5a1.5 1.5 0 0 1 0-3ZM12 6a1 1 0 0 1 1 1v5a1 1 0 0 1-2 0V7a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-warning-filled-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 13.5a1.5 1.5 0 0 0 0 3l.144-.007A1.5 1.5 0 0 0 12 14.5ZM12 6a1 1 0 0 0-1 1v5a1 1 0 0 0 2 0V7a1 1 0 0 0-1-1Z"/></symbol><symbol id="icon-chevron-left-medium" viewBox="0 0 24 24"><path d="M15.7194 3.3054C15.3358 2.90809 14.7027 2.89699 14.3054 3.28061L6.54342 10.7757C6.19804 11.09 6 11.5335 6 12C6 12.4665 6.19804 12.91 6.5218 13.204L14.3054 20.7194C14.7027 21.103 15.3358 21.0919 15.7194 20.6946C16.103 20.2973 16.0919 19.6642 15.6946 19.2806L8.155 12L15.6946 4.71939C16.0614 4.36528 16.099 3.79863 15.8009 3.40105L15.7194 3.3054Z"/></symbol><symbol id="icon-chevron-right-medium" viewBox="0 0 24 24"><path d="M8.28061 3.3054C8.66423 2.90809 9.29729 2.89699 9.6946 3.28061L17.4566 10.7757C17.802 11.09 18 11.5335 18 12C18 12.4665 17.802 12.91 17.4782 13.204L9.6946 20.7194C9.29729 21.103 8.66423 21.0919 8.28061 20.6946C7.89699 20.2973 7.90809 19.6642 8.3054 19.2806L15.845 12L8.3054 4.71939C7.93865 4.36528 7.90098 3.79863 8.19908 3.40105L8.28061 3.3054Z"/></symbol><symbol id="icon-eds-alerts" viewBox="0 0 32 32"><path d="M28 12.667c.736 0 1.333.597 1.333 1.333v13.333A3.333 3.333 0 0 1 26 30.667H6a3.333 3.333 0 0 1-3.333-3.334V14a1.333 1.333 0 1 1 2.666 0v1.252L16 21.769l10.667-6.518V14c0-.736.597-1.333 1.333-1.333Zm-1.333 5.71-9.972 6.094c-.427.26-.963.26-1.39 0l-9.972-6.094v8.956c0 .368.299.667.667.667h20a.667.667 0 0 0 .667-.667v-8.956ZM19.333 12a1.333 1.333 0 1 1 0 2.667h-6.666a1.333 1.333 0 1 1 0-2.667h6.666Zm4-10.667a3.333 3.333 0 0 1 3.334 3.334v6.666a1.333 1.333 0 1 1-2.667 0V4.667A.667.667 0 0 0 23.333 4H8.667A.667.667 0 0 0 8 4.667v6.666a1.333 1.333 0 1 1-2.667 0V4.667a3.333 3.333 0 0 1 3.334-3.334h14.666Zm-4 5.334a1.333 1.333 0 0 1 0 2.666h-6.666a1.333 1.333 0 1 1 0-2.666h6.666Z"/></symbol><symbol id="icon-eds-arrow-up" viewBox="0 0 24 24"><path fill-rule="evenodd" d="m13.002 7.408 4.88 4.88a.99.99 0 0 0 1.32.08l.09-.08c.39-.39.39-1.03 0-1.42l-6.58-6.58a1.01 1.01 0 0 0-1.42 0l-6.58 6.58a1 1 0 0 0-.09 1.32l.08.1a1 1 0 0 0 1.42-.01l4.88-4.87v11.59a.99.99 0 0 0 .88.99l.12.01c.55 0 1-.45 1-1V7.408z" class="layer"/></symbol><symbol id="icon-eds-checklist" viewBox="0 0 32 32"><path d="M19.2 1.333a3.468 3.468 0 0 1 3.381 2.699L24.667 4C26.515 4 28 5.52 28 7.38v19.906c0 1.86-1.485 3.38-3.333 3.38H7.333c-1.848 0-3.333-1.52-3.333-3.38V7.38C4 5.52 5.485 4 7.333 4h2.093A3.468 3.468 0 0 1 12.8 1.333h6.4ZM9.426 6.667H7.333c-.36 0-.666.312-.666.713v19.906c0 .401.305.714.666.714h17.334c.36 0 .666-.313.666-.714V7.38c0-.4-.305-.713-.646-.714l-2.121.033A3.468 3.468 0 0 1 19.2 9.333h-6.4a3.468 3.468 0 0 1-3.374-2.666Zm12.715 5.606c.586.446.7 1.283.253 1.868l-7.111 9.334a1.333 1.333 0 0 1-1.792.306l-3.556-2.333a1.333 1.333 0 1 1 1.463-2.23l2.517 1.651 6.358-8.344a1.333 1.333 0 0 1 1.868-.252ZM19.2 4h-6.4a.8.8 0 0 0-.8.8v1.067a.8.8 0 0 0 .8.8h6.4a.8.8 0 0 0 .8-.8V4.8a.8.8 0 0 0-.8-.8Z"/></symbol><symbol id="icon-eds-citation" viewBox="0 0 36 36"><path d="M23.25 1.5a1.5 1.5 0 0 1 1.06.44l8.25 8.25a1.5 1.5 0 0 1 .44 1.06v19.5c0 2.105-1.645 3.75-3.75 3.75H18a1.5 1.5 0 0 1 0-3h11.25c.448 0 .75-.302.75-.75V11.873L22.628 4.5H8.31a.811.811 0 0 0-.8.68l-.011.13V16.5a1.5 1.5 0 0 1-3 0V5.31A3.81 3.81 0 0 1 8.31 1.5h14.94ZM8.223 20.358a.984.984 0 0 1-.192 1.378l-.048.034c-.54.36-.942.676-1.206.951-.59.614-.885 1.395-.885 2.343.115-.028.288-.042.518-.042.662 0 1.26.237 1.791.711.533.474.799 1.074.799 1.799 0 .753-.259 1.352-.777 1.799-.518.446-1.151.669-1.9.669-1.006 0-1.812-.293-2.417-.878C3.302 28.536 3 27.657 3 26.486c0-1.115.165-2.085.496-2.907.331-.823.734-1.513 1.209-2.071.475-.558.971-.997 1.49-1.318a6.01 6.01 0 0 1 .347-.2 1.321 1.321 0 0 1 1.681.368Zm7.5 0a.984.984 0 0 1-.192 1.378l-.048.034c-.54.36-.942.676-1.206.951-.59.614-.885 1.395-.885 2.343.115-.028.288-.042.518-.042.662 0 1.26.237 1.791.711.533.474.799 1.074.799 1.799 0 .753-.259 1.352-.777 1.799-.518.446-1.151.669-1.9.669-1.006 0-1.812-.293-2.417-.878-.604-.586-.906-1.465-.906-2.636 0-1.115.165-2.085.496-2.907.331-.823.734-1.513 1.209-2.071.475-.558.971-.997 1.49-1.318a6.01 6.01 0 0 1 .347-.2 1.321 1.321 0 0 1 1.681.368Z"/></symbol><symbol id="icon-eds-i-access-indicator" viewBox="0 0 16 16"><circle cx="4.5" cy="11.5" r="3.5" style="fill:currentColor"/><path fill-rule="evenodd" d="M4 3v3a1 1 0 0 1-2 0V2.923C2 1.875 2.84 1 3.909 1h5.909a1 1 0 0 1 .713.298l3.181 3.231a1 1 0 0 1 .288.702v7.846c0 .505-.197.993-.554 1.354a1.902 1.902 0 0 1-1.355.569H10a1 1 0 1 1 0-2h2V5.64L9.4 3H4Z" clip-rule="evenodd" style="fill:#222"/></symbol><symbol id="icon-eds-i-copy-link" viewBox="0 0 24 24"><path fill-rule="evenodd" clip-rule="evenodd" d="M19.4594 8.57015C19.0689 8.17963 19.0689 7.54646 19.4594 7.15594L20.2927 6.32261C20.2927 6.32261 20.2927 6.32261 20.2927 6.32261C21.0528 5.56252 21.0528 4.33019 20.2928 3.57014C19.5327 2.81007 18.3004 2.81007 17.5404 3.57014L16.7071 4.40347C16.3165 4.794 15.6834 4.794 15.2928 4.40348C14.9023 4.01296 14.9023 3.3798 15.2928 2.98927L16.1262 2.15594C17.6673 0.614803 20.1659 0.614803 21.707 2.15593C23.2481 3.69705 23.248 6.19569 21.707 7.7368L20.8737 8.57014C20.4831 8.96067 19.85 8.96067 19.4594 8.57015Z"/><path fill-rule="evenodd" clip-rule="evenodd" d="M18.0944 5.90592C18.4849 6.29643 18.4849 6.9296 18.0944 7.32013L16.4278 8.9868C16.0373 9.37733 15.4041 9.37734 15.0136 8.98682C14.6231 8.59631 14.6231 7.96314 15.0136 7.57261L16.6802 5.90594C17.0707 5.51541 17.7039 5.5154 18.0944 5.90592Z"/><path fill-rule="evenodd" clip-rule="evenodd" d="M13.5113 6.32243C13.9018 6.71295 13.9018 7.34611 13.5113 7.73664L12.678 8.56997C12.678 8.56997 12.678 8.56997 12.678 8.56997C11.9179 9.33006 11.9179 10.5624 12.6779 11.3224C13.438 12.0825 14.6703 12.0825 15.4303 11.3224L16.2636 10.4891C16.6542 10.0986 17.2873 10.0986 17.6779 10.4891C18.0684 10.8796 18.0684 11.5128 17.6779 11.9033L16.8445 12.7366C15.3034 14.2778 12.8048 14.2778 11.2637 12.7366C9.72262 11.1955 9.72266 8.69689 11.2637 7.15578L12.097 6.32244C12.4876 5.93191 13.1207 5.93191 13.5113 6.32243Z"/><path d="M8 20V22H19.4619C20.136 22 20.7822 21.7311 21.2582 21.2529C21.7333 20.7757 22 20.1289 22 19.4549V15C22 14.4477 21.5523 14 21 14C20.4477 14 20 14.4477 20 15V19.4549C20 19.6004 19.9426 19.7397 19.8408 19.842C19.7399 19.9433 19.6037 20 19.4619 20H8Z"/><path d="M4 13H2V19.4619C2 20.136 2.26889 20.7822 2.74705 21.2582C3.22434 21.7333 3.87105 22 4.5451 22H9C9.55228 22 10 21.5523 10 21C10 20.4477 9.55228 20 9 20H4.5451C4.39957 20 4.26028 19.9426 4.15804 19.8408C4.05668 19.7399 4 19.6037 4 19.4619V13Z"/><path d="M4 13H2V4.53808C2 3.86398 2.26889 3.21777 2.74705 2.74178C3.22434 2.26666 3.87105 2 4.5451 2H9C9.55228 2 10 2.44772 10 3C10 3.55228 9.55228 4 9 4H4.5451C4.39957 4 4.26028 4.05743 4.15804 4.15921C4.05668 4.26011 4 4.39633 4 4.53808V13Z"/></symbol><symbol id="icon-eds-i-github-medium" viewBox="0 0 24 24"><path d="M 11.964844 0 C 5.347656 0 0 5.269531 0 11.792969 C 0 17.003906 3.425781 21.417969 8.179688 22.976562 C 8.773438 23.09375 8.992188 22.722656 8.992188 22.410156 C 8.992188 22.136719 8.972656 21.203125 8.972656 20.226562 C 5.644531 20.929688 4.953125 18.820312 4.953125 18.820312 C 4.417969 17.453125 3.625 17.101562 3.625 17.101562 C 2.535156 16.378906 3.703125 16.378906 3.703125 16.378906 C 4.914062 16.457031 5.546875 17.589844 5.546875 17.589844 C 6.617188 19.386719 8.339844 18.878906 9.03125 18.566406 C 9.132812 17.804688 9.449219 17.277344 9.785156 16.984375 C 7.132812 16.710938 4.339844 15.695312 4.339844 11.167969 C 4.339844 9.878906 4.8125 8.824219 5.566406 8.003906 C 5.445312 7.710938 5.03125 6.5 5.683594 4.878906 C 5.683594 4.878906 6.695312 4.566406 8.972656 6.089844 C 9.949219 5.832031 10.953125 5.703125 11.964844 5.699219 C 12.972656 5.699219 14.003906 5.835938 14.957031 6.089844 C 17.234375 4.566406 18.242188 4.878906 18.242188 4.878906 C 18.898438 6.5 18.480469 7.710938 18.363281 8.003906 C 19.136719 8.824219 19.589844 9.878906 19.589844 11.167969 C 19.589844 15.695312 16.796875 16.691406 14.125 16.984375 C 14.558594 17.355469 14.933594 18.058594 14.933594 19.171875 C 14.933594 20.753906 14.914062 22.019531 14.914062 22.410156 C 14.914062 22.722656 15.132812 23.09375 15.726562 22.976562 C 20.480469 21.414062 23.910156 17.003906 23.910156 11.792969 C 23.929688 5.269531 18.558594 0 11.964844 0 Z M 11.964844 0 "/></symbol><symbol id="icon-eds-i-institution-medium" viewBox="0 0 24 24"><g><path fill-rule="evenodd" clip-rule="evenodd" d="M11.9967 1C11.6364 1 11.279 1.0898 10.961 1.2646C10.9318 1.28061 10.9035 1.29806 10.8761 1.31689L2.79765 6.87C2.46776 7.08001 2.20618 7.38466 2.07836 7.76668C1.94823 8.15561 1.98027 8.55648 2.12665 8.90067C2.42086 9.59246 3.12798 10 3.90107 10H4.99994V16H4.49994C3.11923 16 1.99994 17.1193 1.99994 18.5V19.5C1.99994 20.8807 3.11923 22 4.49994 22H19.4999C20.8807 22 21.9999 20.8807 21.9999 19.5V18.5C21.9999 17.1193 20.8807 16 19.4999 16H18.9999V10H20.0922C20.8653 10 21.5725 9.59252 21.8667 8.90065C22.0131 8.55642 22.0451 8.15553 21.9149 7.7666C21.7871 7.38459 21.5255 7.07997 21.1956 6.86998L13.1172 1.31689C13.0898 1.29806 13.0615 1.28061 13.0324 1.2646C12.7143 1.0898 12.357 1 11.9967 1ZM4.6844 8L11.9472 3.00755C11.9616 3.00295 11.9783 3 11.9967 3C12.015 3 12.0318 3.00295 12.0461 3.00755L19.3089 8H4.6844ZM16.9999 16V10H14.9999V16H16.9999ZM12.9999 16V10H10.9999V16H12.9999ZM8.99994 16V10H6.99994V16H8.99994ZM3.99994 18.5C3.99994 18.2239 4.2238 18 4.49994 18H19.4999C19.7761 18 19.9999 18.2239 19.9999 18.5V19.5C19.9999 19.7761 19.7761 20 19.4999 20H4.49994C4.2238 20 3.99994 19.7761 3.99994 19.5V18.5Z"/></g></symbol><symbol id="icon-eds-i-limited-access" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 3v3a1 1 0 0 1-2 0V2.923C2 1.875 2.84 1 3.909 1h5.909a1 1 0 0 1 .713.298l3.181 3.231a1 1 0 0 1 .288.702V6a1 1 0 1 1-2 0v-.36L9.4 3H4ZM3 8a1 1 0 0 1 1 1v1a1 1 0 1 1-2 0V9a1 1 0 0 1 1-1Zm10 0a1 1 0 0 1 1 1v1a1 1 0 1 1-2 0V9a1 1 0 0 1 1-1Zm-3.5 6a1 1 0 0 1-1 1h-1a1 1 0 1 1 0-2h1a1 1 0 0 1 1 1Zm2.441-1a1 1 0 0 1 2 0c0 .73-.246 1.306-.706 1.664a1.61 1.61 0 0 1-.876.334l-.032.002H11.5a1 1 0 1 1 0-2h.441ZM4 13a1 1 0 0 0-2 0c0 .73.247 1.306.706 1.664a1.609 1.609 0 0 0 .876.334l.032.002H4.5a1 1 0 1 0 0-2H4Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-search-category-medium" viewBox="0 0 32 32"><path fill-rule="evenodd" d="M2 5.306A3.306 3.306 0 0 1 5.306 2h5.833a3.306 3.306 0 0 1 3.306 3.306v5.833a3.306 3.306 0 0 1-3.306 3.305H5.306A3.306 3.306 0 0 1 2 11.14V5.306Zm3.306-.584a.583.583 0 0 0-.584.584v5.833c0 .322.261.583.584.583h5.833a.583.583 0 0 0 .583-.583V5.306a.583.583 0 0 0-.583-.584H5.306Zm15.555 8.945a7.194 7.194 0 1 0 4.034 13.153l2.781 2.781a1.361 1.361 0 1 0 1.925-1.925l-2.781-2.781a7.194 7.194 0 0 0-5.958-11.228Zm3.173 10.346a4.472 4.472 0 1 0-.021.021l.01-.01.011-.011Zm-5.117-19.29a.583.583 0 0 0-.584.583v5.833a1.361 1.361 0 0 1-2.722 0V5.306A3.306 3.306 0 0 1 18.917 2h5.833a3.306 3.306 0 0 1 3.306 3.306v5.833c0 .6-.161 1.166-.443 1.654a1.361 1.361 0 1 1-2.357-1.363.575.575 0 0 0 .078-.291V5.306a.583.583 0 0 0-.584-.584h-5.833ZM2 18.916a3.306 3.306 0 0 1 3.306-3.306h5.833a1.361 1.361 0 1 1 0 2.722H5.306a.583.583 0 0 0-.584.584v5.833c0 .322.261.583.584.583h5.833a.574.574 0 0 0 .29-.077 1.361 1.361 0 1 1 1.364 2.356 3.296 3.296 0 0 1-1.654.444H5.306A3.306 3.306 0 0 1 2 24.75v-5.833Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-subjects-medium" viewBox="0 0 24 24"><g id="icon-subjects-copy" stroke="none" stroke-width="1" fill-rule="evenodd"><path d="M13.3846154,2 C14.7015971,2 15.7692308,3.06762994 15.7692308,4.38461538 L15.7692308,7.15384615 C15.7692308,8.47082629 14.7015955,9.53846154 13.3846154,9.53846154 L13.1038388,9.53925278 C13.2061091,9.85347965 13.3815528,10.1423885 13.6195822,10.3804178 C13.9722182,10.7330539 14.436524,10.9483278 14.9293854,10.9918129 L15.1153846,11 C16.2068332,11 17.2535347,11.433562 18.0254647,12.2054189 C18.6411944,12.8212361 19.0416785,13.6120766 19.1784166,14.4609738 L19.6153846,14.4615385 C20.932386,14.4615385 22,15.5291672 22,16.8461538 L22,19.6153846 C22,20.9323924 20.9323924,22 19.6153846,22 L16.8461538,22 C15.5291672,22 14.4615385,20.932386 14.4615385,19.6153846 L14.4615385,16.8461538 C14.4615385,15.5291737 15.5291737,14.4615385 16.8461538,14.4615385 L17.126925,14.460779 C17.0246537,14.1465537 16.8492179,13.857633 16.6112344,13.6196157 C16.2144418,13.2228606 15.6764136,13 15.1153846,13 C14.0239122,13 12.9771569,12.5664197 12.2053686,11.7946314 C12.1335167,11.7227795 12.0645962,11.6485444 11.9986839,11.5721119 C11.9354038,11.6485444 11.8664833,11.7227795 11.7946314,11.7946314 C11.0228431,12.5664197 9.97608778,13 8.88461538,13 C8.323576,13 7.78552852,13.2228666 7.38881294,13.6195822 C7.15078359,13.8576115 6.97533988,14.1465203 6.8730696,14.4607472 L7.15384615,14.4615385 C8.47082629,14.4615385 9.53846154,15.5291737 9.53846154,16.8461538 L9.53846154,19.6153846 C9.53846154,20.932386 8.47083276,22 7.15384615,22 L4.38461538,22 C3.06762347,22 2,20.9323876 2,19.6153846 L2,16.8461538 C2,15.5291721 3.06762994,14.4615385 4.38461538,14.4615385 L4.8215823,14.4609378 C4.95831893,13.6120029 5.3588057,12.8211623 5.97459937,12.2053686 C6.69125996,11.488708 7.64500941,11.0636656 8.6514968,11.0066017 L8.88461538,11 C9.44565477,11 9.98370225,10.7771334 10.3804178,10.3804178 C10.6184472,10.1423885 10.7938909,9.85347965 10.8961612,9.53925278 L10.6153846,9.53846154 C9.29840448,9.53846154 8.23076923,8.47082629 8.23076923,7.15384615 L8.23076923,4.38461538 C8.23076923,3.06762994 9.29840286,2 10.6153846,2 L13.3846154,2 Z M7.15384615,16.4615385 L4.38461538,16.4615385 C4.17220099,16.4615385 4,16.63374 4,16.8461538 L4,19.6153846 C4,19.8278134 4.17218833,20 4.38461538,20 L7.15384615,20 C7.36626945,20 7.53846154,19.8278103 7.53846154,19.6153846 L7.53846154,16.8461538 C7.53846154,16.6337432 7.36625679,16.4615385 7.15384615,16.4615385 Z M19.6153846,16.4615385 L16.8461538,16.4615385 C16.6337432,16.4615385 16.4615385,16.6337432 16.4615385,16.8461538 L16.4615385,19.6153846 C16.4615385,19.8278103 16.6337306,20 16.8461538,20 L19.6153846,20 C19.8278229,20 20,19.8278229 20,19.6153846 L20,16.8461538 C20,16.6337306 19.8278103,16.4615385 19.6153846,16.4615385 Z M13.3846154,4 L10.6153846,4 C10.4029708,4 10.2307692,4.17220099 10.2307692,4.38461538 L10.2307692,7.15384615 C10.2307692,7.36625679 10.402974,7.53846154 10.6153846,7.53846154 L13.3846154,7.53846154 C13.597026,7.53846154 13.7692308,7.36625679 13.7692308,7.15384615 L13.7692308,4.38461538 C13.7692308,4.17220099 13.5970292,4 13.3846154,4 Z" id="Shape" fill-rule="nonzero"/></g></symbol><symbol id="icon-eds-small-arrow-left" viewBox="0 0 16 17"><path stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" d="M14 8.092H2m0 0L8 2M2 8.092l6 6.035"/></symbol><symbol id="icon-eds-small-arrow-right" viewBox="0 0 16 16"><g fill-rule="evenodd" stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="2"><path d="M2 8.092h12M8 2l6 6.092M8 14.127l6-6.035"/></g></symbol><symbol id="icon-orcid-logo" viewBox="0 0 40 40"><path fill-rule="evenodd" d="M12.281 10.453c.875 0 1.578-.719 1.578-1.578 0-.86-.703-1.578-1.578-1.578-.875 0-1.578.703-1.578 1.578 0 .86.703 1.578 1.578 1.578Zm-1.203 18.641h2.406V12.359h-2.406v16.735Z"/><path fill-rule="evenodd" d="M17.016 12.36h6.5c6.187 0 8.906 4.421 8.906 8.374 0 4.297-3.36 8.375-8.875 8.375h-6.531V12.36Zm6.234 14.578h-3.828V14.53h3.703c4.688 0 6.828 2.844 6.828 6.203 0 2.063-1.25 6.203-6.703 6.203Z" clip-rule="evenodd"/></symbol></svg> </div> <a class="c-skip-link" href="#main">Skip to main content</a> <header class="eds-c-header" data-eds-c-header> <div class="eds-c-header__container" data-eds-c-header-expander-anchor> <div class="eds-c-header__brand"> <a href="https://link.springer.com" data-test=springerlink-logo data-track="click_imprint_logo" data-track-context="unified header" data-track-action="click logo link" data-track-category="unified header" data-track-label="link" > <img src="/oscar-static/images/darwin/header/img/logo-springer-nature-link-3149409f62.svg" alt="Springer Nature Link"> </a> </div> <a class="c-header__link eds-c-header__link" id="identity-account-widget" data-track="click_login" data-track-context="header" href='https://idp.springer.com/auth/personal/springernature?redirect_uri=https://link.springer.com/article/10.1007/s40062-016-0139-x?'><span class="eds-c-header__widget-fragment-title">Log in</span></a> </div> <nav class="eds-c-header__nav" aria-label="header navigation"> <div class="eds-c-header__nav-container"> <div class="eds-c-header__item eds-c-header__item--menu"> <a href="#eds-c-header-nav" class="eds-c-header__link" data-eds-c-header-expander> <svg class="eds-c-header__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-menu-medium"></use> </svg><span>Menu</span> </a> </div> <div class="eds-c-header__item eds-c-header__item--inline-links"> <a class="eds-c-header__link" href="https://link.springer.com/journals/" data-track="nav_find_a_journal" data-track-context="unified header" data-track-action="click find a journal" data-track-category="unified header" data-track-label="link" > Find a journal </a> <a class="eds-c-header__link" href="https://www.springernature.com/gp/authors" data-track="nav_how_to_publish" data-track-context="unified header" data-track-action="click publish with us link" data-track-category="unified header" data-track-label="link" > Publish with us </a> <a class="eds-c-header__link" href="https://link.springernature.com/home/" data-track="nav_track_your_research" data-track-context="unified header" data-track-action="click track your research" data-track-category="unified header" data-track-label="link" > Track your research </a> </div> <div class="eds-c-header__link-container"> <div class="eds-c-header__item eds-c-header__item--divider"> <a href="#eds-c-header-popup-search" class="eds-c-header__link" data-eds-c-header-expander data-eds-c-header-test-search-btn> <svg class="eds-c-header__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-search-medium"></use> </svg><span>Search</span> </a> </div> <div id="ecommerce-header-cart-icon-link" class="eds-c-header__item ecommerce-cart" style="display:inline-block"> <a class="eds-c-header__link" href="https://order.springer.com/public/cart" style="appearance:none;border:none;background:none;color:inherit;position:relative"> <svg id="eds-i-cart" class="eds-c-header__icon" xmlns="http://www.w3.org/2000/svg" height="24" width="24" viewBox="0 0 24 24" aria-hidden="true" focusable="false"> <path fill="currentColor" fill-rule="nonzero" d="M2 1a1 1 0 0 0 0 2l1.659.001 2.257 12.808a2.599 2.599 0 0 0 2.435 2.185l.167.004 9.976-.001a2.613 2.613 0 0 0 2.61-1.748l.03-.106 1.755-7.82.032-.107a2.546 2.546 0 0 0-.311-1.986l-.108-.157a2.604 2.604 0 0 0-2.197-1.076L6.042 5l-.56-3.17a1 1 0 0 0-.864-.82l-.12-.007L2.001 1ZM20.35 6.996a.63.63 0 0 1 .54.26.55.55 0 0 1 .082.505l-.028.1L19.2 15.63l-.022.05c-.094.177-.282.299-.526.317l-10.145.002a.61.61 0 0 1-.618-.515L6.394 6.999l13.955-.003ZM18 19a2 2 0 1 0 0 4 2 2 0 0 0 0-4ZM8 19a2 2 0 1 0 0 4 2 2 0 0 0 0-4Z"></path> </svg><span>Cart</span><span class="cart-info" style="display:none;position:absolute;top:10px;right:45px;background-color:#C65301;color:#fff;width:18px;height:18px;font-size:11px;border-radius:50%;line-height:17.5px;text-align:center"></span></a> <script>(function () { var exports = {}; if (window.fetch) { "use strict"; Object.defineProperty(exports, "__esModule", { value: true }); exports.headerWidgetClientInit = void 0; var headerWidgetClientInit = function (getCartInfo) { document.body.addEventListener("updatedCart", function () { updateCartIcon(); }, false); return updateCartIcon(); function updateCartIcon() { return getCartInfo() .then(function (res) { return res.json(); }) .then(refreshCartState) .catch(function (_) { }); } function refreshCartState(json) { var indicator = document.querySelector("#ecommerce-header-cart-icon-link .cart-info"); /* istanbul ignore else */ if (indicator && json.itemCount) { indicator.style.display = 'block'; indicator.textContent = json.itemCount > 9 ? '9+' : json.itemCount.toString(); var moreThanOneItem = json.itemCount > 1; indicator.setAttribute('title', "there ".concat(moreThanOneItem ? "are" : "is", " ").concat(json.itemCount, " item").concat(moreThanOneItem ? "s" : "", " in your cart")); } return json; } }; exports.headerWidgetClientInit = headerWidgetClientInit; headerWidgetClientInit( function () { return window.fetch("https://cart.springer.com/cart-info", { credentials: "include", headers: { Accept: "application/json" } }) } ) }})()</script> </div> </div> </div> </nav> </header> <article lang="en" id="main" class="app-masthead__colour-6"> <section class="app-masthead " aria-label="article masthead"> <div class="app-masthead__container"> <div class="app-article-masthead u-sans-serif js-context-bar-sticky-point-masthead" data-track-component="article" data-test="masthead-component"> <div class="app-article-masthead__info"> <nav aria-label="breadcrumbs" data-test="breadcrumbs"> <ol class="c-breadcrumbs c-breadcrumbs--contrast" itemscope itemtype="https://schema.org/BreadcrumbList"> <li class="c-breadcrumbs__item" id="breadcrumb0" itemprop="itemListElement" itemscope="" itemtype="https://schema.org/ListItem"> <a href="/" class="c-breadcrumbs__link" itemprop="item" data-track="click_breadcrumb" data-track-context="article page" data-track-category="article" data-track-action="breadcrumbs" data-track-label="breadcrumb1"><span itemprop="name">Home</span></a><meta itemprop="position" content="1"> <svg class="c-breadcrumbs__chevron" role="img" aria-hidden="true" focusable="false" width="10" height="10" viewBox="0 0 10 10"> <path d="m5.96738168 4.70639573 2.39518594-2.41447274c.37913917-.38219212.98637524-.38972225 1.35419292-.01894278.37750606.38054586.37784436.99719163-.00013556 1.37821513l-4.03074001 4.06319683c-.37758093.38062133-.98937525.38100976-1.367372-.00003075l-4.03091981-4.06337806c-.37759778-.38063832-.38381821-.99150444-.01600053-1.3622839.37750607-.38054587.98772445-.38240057 1.37006824.00302197l2.39538588 2.4146743.96295325.98624457z" fill-rule="evenodd" transform="matrix(0 -1 1 0 0 10)"/> </svg> </li> <li class="c-breadcrumbs__item" id="breadcrumb1" itemprop="itemListElement" itemscope="" itemtype="https://schema.org/ListItem"> <a href="/journal/40062" class="c-breadcrumbs__link" itemprop="item" data-track="click_breadcrumb" data-track-context="article page" data-track-category="article" data-track-action="breadcrumbs" data-track-label="breadcrumb2"><span itemprop="name">Journal of Homotopy and Related Structures</span></a><meta itemprop="position" content="2"> <svg class="c-breadcrumbs__chevron" role="img" aria-hidden="true" focusable="false" width="10" height="10" viewBox="0 0 10 10"> <path d="m5.96738168 4.70639573 2.39518594-2.41447274c.37913917-.38219212.98637524-.38972225 1.35419292-.01894278.37750606.38054586.37784436.99719163-.00013556 1.37821513l-4.03074001 4.06319683c-.37758093.38062133-.98937525.38100976-1.367372-.00003075l-4.03091981-4.06337806c-.37759778-.38063832-.38381821-.99150444-.01600053-1.3622839.37750607-.38054587.98772445-.38240057 1.37006824.00302197l2.39538588 2.4146743.96295325.98624457z" fill-rule="evenodd" transform="matrix(0 -1 1 0 0 10)"/> </svg> </li> <li class="c-breadcrumbs__item" id="breadcrumb2" itemprop="itemListElement" itemscope="" itemtype="https://schema.org/ListItem"> <span itemprop="name">Article</span><meta itemprop="position" content="3"> </li> </ol> </nav> <h1 class="c-article-title" data-test="article-title" data-article-title="">Frames in cofibration categories</h1> <ul class="c-article-identifiers"> <li class="c-article-identifiers__item"> Published: <time datetime="2016-07-13">13 July 2016</time> </li> </ul> <ul class="c-article-identifiers c-article-identifiers--cite-list"> <li class="c-article-identifiers__item"> <span data-test="journal-volume">Volume 12</span>, pages 577–616, (<span data-test="article-publication-year">2017</span>) </li> <li class="c-article-identifiers__item c-article-identifiers__item--cite"> <a href="#citeas" data-track="click" data-track-action="cite this article" data-track-category="article body" data-track-label="link">Cite this article</a> </li> </ul> <div class="app-article-masthead__buttons" data-test="download-article-link-wrapper" data-track-context="masthead"> <div class="c-pdf-container"> <div class="c-pdf-download u-clear-both u-mb-16"> <a href="/content/pdf/10.1007/s40062-016-0139-x.pdf" class="u-button u-button--full-width u-button--primary u-justify-content-space-between c-pdf-download__link" data-article-pdf="true" data-readcube-pdf-url="true" data-test="pdf-link" data-draft-ignore="true" data-track="content_download" data-track-type="article pdf download" data-track-action="download pdf" data-track-label="button" data-track-external download> <span class="c-pdf-download__text">Download PDF</span> <svg aria-hidden="true" focusable="false" width="16" height="16" class="u-icon"><use xlink:href="#icon-eds-i-download-medium"/></svg> </a> </div> </div> </div> </div> <div class="app-article-masthead__brand"> <a href="/journal/40062" class="app-article-masthead__journal-link" data-track="click_journal_home" data-track-action="journal homepage" data-track-context="article page" data-track-label="link"> <picture> <source type="image/webp" media="(min-width: 768px)" width="120" height="159" srcset="https://media.springernature.com/w120/springer-static/cover-hires/journal/40062?as=webp, https://media.springernature.com/w316/springer-static/cover-hires/journal/40062?as=webp 2x"> <img width="72" height="95" src="https://media.springernature.com/w72/springer-static/cover-hires/journal/40062?as=webp" srcset="https://media.springernature.com/w144/springer-static/cover-hires/journal/40062?as=webp 2x" alt=""> </picture> <span class="app-article-masthead__journal-title">Journal of Homotopy and Related Structures</span> </a> <a href="https://link.springer.com/journal/40062/aims-and-scope" class="app-article-masthead__submission-link" data-track="click_aims_and_scope" data-track-action="aims and scope" data-track-context="article page" data-track-label="link"> Aims and scope <svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-arrow-right-medium"></use></svg> </a> <a href="https://tcms.org.ge/Journals/submission/paper_submit.php" class="app-article-masthead__submission-link" data-track="click_submit_manuscript" data-track-context="article masthead on springerlink article page" data-track-action="submit manuscript" data-track-label="link"> Submit manuscript <svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-arrow-right-medium"></use></svg> </a> </div> </div> </div> </section> <div class="c-article-main u-container u-mt-24 u-mb-32 l-with-sidebar" id="main-content" data-component="article-container"> <main class="u-serif js-main-column" data-track-component="article body"> <div class="c-context-bar u-hide" data-test="context-bar" data-context-bar aria-hidden="true"> <div class="c-context-bar__container u-container"> <div class="c-context-bar__title"> Frames in cofibration categories </div> <div data-test="inCoD" data-track-context="sticky banner"> <div class="c-pdf-container"> <div class="c-pdf-download u-clear-both u-mb-16"> <a href="/content/pdf/10.1007/s40062-016-0139-x.pdf" class="u-button u-button--full-width u-button--primary u-justify-content-space-between c-pdf-download__link" data-article-pdf="true" data-readcube-pdf-url="true" data-test="pdf-link" data-draft-ignore="true" data-track="content_download" data-track-type="article pdf download" data-track-action="download pdf" data-track-label="button" data-track-external download> <span class="c-pdf-download__text">Download PDF</span> <svg aria-hidden="true" focusable="false" width="16" height="16" class="u-icon"><use xlink:href="#icon-eds-i-download-medium"/></svg> </a> </div> </div> </div> </div> </div> <div class="c-article-header"> <header> <ul class="c-article-author-list c-article-author-list--short" data-test="authors-list" data-component-authors-activator="authors-list"><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Karol-Szumi_o-Aff1" data-author-popup="auth-Karol-Szumi_o-Aff1" data-author-search="Szumiło, Karol" data-corresp-id="c1">Karol Szumiło<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><sup class="u-js-hide"><a href="#Aff1">1</a></sup> </li></ul> <div data-test="article-metrics"> <ul class="app-article-metrics-bar u-list-reset"> <li class="app-article-metrics-bar__item"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-accesses-medium"></use> </svg>1681 <span class="app-article-metrics-bar__label">Accesses</span></p> </li> <li class="app-article-metrics-bar__item app-article-metrics-bar__item--metrics"> <p class="app-article-metrics-bar__details"><a href="/article/10.1007/s40062-016-0139-x/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Explore all metrics <svg class="u-icon app-article-metrics-bar__arrow-icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-arrow-right-medium"></use> </svg></a></p> </li> </ul> </div> <div class="u-mt-32"> </div> </header> </div> <div data-article-body="true" data-track-component="article body" class="c-article-body"> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>We introduce the quasicategory of frames of a cofibration category, i.e. a new model of the <span class="mathjax-tex">$(\infty ,1)$</span>-category associated with a cofibration category.</p></div></div></section> <div data-test="cobranding-download"> </div> <section aria-labelledby="inline-recommendations" data-title="Inline Recommendations" class="c-article-recommendations" data-track-component="inline-recommendations"> <h3 class="c-article-recommendations-title" id="inline-recommendations">Similar content being viewed by others</h3> <div class="c-article-recommendations-list"> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3Aplaceholder%2Fimages/placeholder-figure-springernature.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/s10485-015-9422-y?fromPaywallRec=false" data-track="select_recommendations_1" data-track-context="inline recommendations" data-track-action="click recommendations inline - 1" data-track-label="10.1007/s10485-015-9422-y">Quasicategories of Frames of Cofibration Categories </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">20 February 2016</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3Aplaceholder%2Fimages/placeholder-figure-springernature.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/s00032-023-00383-4?fromPaywallRec=false" data-track="select_recommendations_2" data-track-context="inline recommendations" data-track-action="click recommendations inline - 2" data-track-label="10.1007/s00032-023-00383-4">The Third Cohomology 2-Group </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">19 July 2023</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w92h120/springer-static/cover-hires/book/978-3-319-72299-3?as=webp" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/978-3-319-72299-3_2?fromPaywallRec=false" data-track="select_recommendations_3" data-track-context="inline recommendations" data-track-action="click recommendations inline - 3" data-track-label="10.1007/978-3-319-72299-3_2">Fibrations in ∞-Category Theory </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Chapter</span> <span class="c-article-meta-recommendations__date">© 2018</span> </div> </div> </article> </div> </div> </section> <script> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ recommendations: { recommender: 'semantic', model: 'specter', policy_id: 'NA', timestamp: 1740128812, embedded_user: 'null' } }); </script> <div class="app-card-service" data-test="article-checklist-banner"> <div> <a class="app-card-service__link" data-track="click_presubmission_checklist" data-track-context="article page top of reading companion" data-track-category="pre-submission-checklist" data-track-action="clicked article page checklist banner test 2 old version" data-track-label="link" href="https://beta.springernature.com/pre-submission?journalId=40062" data-test="article-checklist-banner-link"> <span class="app-card-service__link-text">Use our pre-submission checklist</span> <svg class="app-card-service__link-icon" aria-hidden="true" focusable="false"><use xlink:href="#icon-eds-i-arrow-right-small"></use></svg> </a> <p class="app-card-service__description">Avoid common mistakes on your manuscript.</p> </div> <div class="app-card-service__icon-container"> <svg class="app-card-service__icon" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-clipboard-check-medium"></use> </svg> </div> </div> <div class="main-content"> <section data-title="Introduction"><div class="c-article-section" id="Sec1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec1"><span class="c-article-section__title-number">1 </span>Introduction</h2><div class="c-article-section__content" id="Sec1-content"><p>There are a number of ways of constructing a quasicategory associated with a cofibration category. For example one could take the derived homotopy coherent nerve [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Cordier, J.M.: Sur la notion de diagramme homotopiquement cohérent. Cahiers Topologie Géom. Différentielle 23(1), 93–112 (1982) (French). Third Colloquium on Categories, Part VI (Amiens, 1980)" href="/article/10.1007/s40062-016-0139-x#ref-CR4" id="ref-link-section-d81634044e322">4</a>] of its hammock localization [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="Dwyer, W.G., Kan, D.M.: Simplicial localizations of categories. J. Pure Appl. Algebra 17(3), 267–284 (1980)" href="/article/10.1007/s40062-016-0139-x#ref-CR5" id="ref-link-section-d81634044e325">5</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="Dwyer, W.G., Kan, D.M.: Calculating simplicial localizations. J. Pure Appl. Algebra 18(1), 17–35 (1980)" href="/article/10.1007/s40062-016-0139-x#ref-CR6" id="ref-link-section-d81634044e328">6</a>] or apply the derived functor of the Quillen functor <span class="mathjax-tex">$i_1^*$</span> of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="Joyal, A., Tierney, M.: Quasi-categories vs Segal spaces. Categories in algebra, geometry and mathematical physics, Contemp. Math., vol. 431, pp. 277–326. Amer. Math. Soc., Providence, RI (2007)" href="/article/10.1007/s40062-016-0139-x#ref-CR11" id="ref-link-section-d81634044e357">11</a>] to its classification diagram [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="Rezk, C.: A model for the homotopy theory of homotopy theory. Trans. Amer. Math. Soc. 353(3), 973–1007 (2001) (electronic)" href="/article/10.1007/s40062-016-0139-x#ref-CR15" id="ref-link-section-d81634044e361">15</a>]. Of course these constructions only depend on the weak equivalences and not on cofibrations.</p><p>In this paper we introduce a new one called the <i>quasicategory of frames</i> of a cofibration category. It has a number of convenient features compared to the constructions above. It directly yields a quasicategory so that no fibrant replacement in the Joyal model structure (or any of related model categories) is necessary. It does not rely on simplicial enrichment. It takes advantage of the structure of a cofibration category in such a way that homotopy colimits constructed using methods of homotopical algebra can be quite directly translated into quasicategorical colimits.</p><p>This paper is the second in the series of three that summarize the results of the author’s thesis [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Szumiło, K.: Two Models for the Homotopy Theory of Cocomplete Homotopy Theories. Ph.D. Thesis, Rheinische Friedrich-Wilhelms-Universität Bonn (2014). 
 http://hss.ulb.uni-bonn.de/2014/3692/3692.htm
 
 
 " href="/article/10.1007/s40062-016-0139-x#ref-CR19" id="ref-link-section-d81634044e373">19</a>] (see also [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title="Szumiło, K.: Two Models for the Homotopy Theory of Cocomplete Homotopy Theories (2014). 
 arXiv:1411.0303
 
 
 " href="/article/10.1007/s40062-016-0139-x#ref-CR20" id="ref-link-section-d81634044e376">20</a>] for a slightly edited version). The first one [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Szumiło, K.: Homotopy theory of cofibration categories. Homol. Homotop. Appl (to appear)" href="/article/10.1007/s40062-016-0139-x#ref-CR21" id="ref-link-section-d81634044e379">21</a>] constructs a fibration category of cofibration categories which provides a convenient framework for the homotopy theory of cofibration categories. In particular, the notion of a fibration of cofibration categories introduced there is a crucial tool in the present paper. The third one [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="Szumiło, K.: Homotopy theory of cocomplete quasicategories. Algebraic Geom Topol (to appear)" href="/article/10.1007/s40062-016-0139-x#ref-CR22" id="ref-link-section-d81634044e382">22</a>] shows that it is possible to reconstruct a cofibration category from a cocomplete quasicategory in a way that establishes an equivalence between the homotopy theory of cofibration categories and the homotopy theory of cocomplete quasicategories. Moreover, in a paper joint with Chris Kapulkin [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title="Kapulkin, K., Szumiło, K.: Quasicategories of frames of cofibration categories. Appl. Categ. Struct., 1–25 (2016)" href="/article/10.1007/s40062-016-0139-x#ref-CR13" id="ref-link-section-d81634044e385">13</a>] we prove that the quasicategory of frames models the simplicial localization of a cofibration category.</p><p>As an example of an application of this construction, it can be shown that the simplicial localization of any categorical model of dependent type theory is a locally cartesian closed quasicategory [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="Kapulkin, K.: Locally Cartesian Closed Quasicategories From Type Theory. 
 arXiv:1507.02648
 
 
 " href="/article/10.1007/s40062-016-0139-x#ref-CR12" id="ref-link-section-d81634044e391">12</a>]. This problem has proven difficult when working with known models of simplicial localization. However, every categorical model of type theory is a fibration category [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="Avigad, J., Kapulkin, K., Lumsdaine, P.L.: Homotopy limits in type theory. Math. Struct. Comput. Sci. 25(5), 1040–1070 (2015)" href="/article/10.1007/s40062-016-0139-x#ref-CR1" id="ref-link-section-d81634044e394">1</a>, Theorem 3.2.5] and hence, by results of the present paper, its localization is a quasicategory with finite limits and our methods can also be used to prove that it is locally cartesian closed. This result can be seen as a step towards describing internal languages of higher categories.</p><p>We start with Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s40062-016-0139-x#Sec2">1</a> where we prove a number of preliminary results about diagrams in cofibration categories and fibrations between fibration categories. This section builds directly on [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Szumiło, K.: Homotopy theory of cofibration categories. Homol. Homotop. Appl (to appear)" href="/article/10.1007/s40062-016-0139-x#ref-CR21" id="ref-link-section-d81634044e404">21</a>]. In Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s40062-016-0139-x#Sec3">2</a> we construct a functor from cofibration categories to cocomplete quasicategories. To each cofibration category <span class="mathjax-tex">$\mathcal {C}$</span> we associate a nerve-like simplicial set denoted by <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> and called the <i>quasicategory of frames in</i> <span class="mathjax-tex">$\mathcal {C}$</span> (the letter <span class="mathjax-tex">$\mathrm {f}$</span> in <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}$</span> stands either for <i>frames</i> since those are the objects in <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> or for <i>fractions</i> since the morphisms in <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> are certain generalizations of left fractions). In Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s40062-016-0139-x#Sec4">3</a> we show that <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> is a quasicategory an in Sects. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s40062-016-0139-x#Sec5">4</a> and <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s40062-016-0139-x#Sec6">5</a> that it is cocomplete.</p><p>Our results are parametrized by a regular cardinal <span class="mathjax-tex">$\kappa $</span>. We fix such <span class="mathjax-tex">$\kappa $</span> and show that if <span class="mathjax-tex">$\mathcal {C}$</span> is a <span class="mathjax-tex">$\kappa $</span>-cocomplete cofibration category, then <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> is a <span class="mathjax-tex">$\kappa $</span>-cocomplete quasicategory. In Sects. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s40062-016-0139-x#Sec3">2</a> and <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s40062-016-0139-x#Sec4">3</a>, to simplify the notation, we take <span class="mathjax-tex">$\kappa = \aleph _0$</span> since the results and methods sections do not depend on <span class="mathjax-tex">$\kappa $</span>. However, the discussion of cocompleteness splits into two cases. The (easier) case of <span class="mathjax-tex">$\kappa > \aleph _0$</span> is dealt with in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s40062-016-0139-x#Sec5">4</a> and the case of <span class="mathjax-tex">$\kappa = \aleph _0$</span> in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s40062-016-0139-x#Sec6">5</a>.</p></div></div></section><section data-title="Cofibration categories of diagrams"><div class="c-article-section" id="Sec2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec2"><span class="c-article-section__title-number">2 </span>Cofibration categories of diagrams</h2><div class="c-article-section__content" id="Sec2-content"><p>Our results are based on the techniques of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Szumiło, K.: Homotopy theory of cofibration categories. Homol. Homotop. Appl (to appear)" href="/article/10.1007/s40062-016-0139-x#ref-CR21" id="ref-link-section-d81634044e917">21</a>] and we start by summarizing the contents of this paper. The central notion is that of <i>cofibration categories</i> which are slightly modified duals of Brown’s <i>categories of fibrant objects</i> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="Brown, K.S.: Abstract homotopy theory and generalized sheaf cohomology. Trans. Am. Math. Soc. 186, 419–458 (1973)" href="/article/10.1007/s40062-016-0139-x#ref-CR3" id="ref-link-section-d81634044e926">3</a>].</p> <h3 class="c-article__sub-heading" id="FPar1">Definition 1.1</h3> <p>[<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Szumiło, K.: Homotopy theory of cofibration categories. Homol. Homotop. Appl (to appear)" href="/article/10.1007/s40062-016-0139-x#ref-CR21" id="ref-link-section-d81634044e936">21</a>, Definition 1.1] A <i>cofibration category</i> is a category <span class="mathjax-tex">$\mathcal {C}$</span> equipped with two subcategories: the subcategory of <i>weak equivalences</i> (denoted by <span class="mathjax-tex">$\mathop {\rightarrow }\limits ^{\sim }$</span>) and the subcategory of <i>cofibrations</i> (denoted by <span class="mathjax-tex">$\rightarrowtail $</span>) such that the following axioms are satisfied (here, an <i>acyclic cofibration</i> is a morphism that is both a weak equivalence and a cofibration).</p><ul class="u-list-style-none"> <li> <p>(C0) Weak equivalences satisfy the 2-out-of-6 property, i.e. if </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-a"><figure><div class="c-article-section__figure-content" id="Figa"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figa_HTML.gif?as=webp"><img aria-describedby="Figa" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figa_HTML.gif" alt="figure a" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-a-desc"></div></div></figure></div><p> are morphisms of <span class="mathjax-tex">$\mathcal {C}$</span> such that both <i>gf</i> and <i>hg</i> are weak equivalences, then so are <i>f</i>, <i>g</i> and <i>h</i> (and thus also <i>hg</i> <i>f</i>).</p> </li> <li> <p>(C1) Every isomorphism of <span class="mathjax-tex">$\mathcal {C}$</span> is an acyclic cofibration.</p> </li> <li> <p>(C2) An initial object exists in <span class="mathjax-tex">$\mathcal {C}$</span>.</p> </li> <li> <p>(C3) Every object <i>X</i> of <span class="mathjax-tex">$\mathcal {C}$</span> is cofibrant, i.e. if <span class="mathjax-tex">$0$</span> is the initial object of <span class="mathjax-tex">$\mathcal {C}$</span>, then the unique morphism <span class="mathjax-tex">$0\rightarrow X$</span> is a cofibration.</p> </li> <li> <p>(C4) Cofibrations are stable under pushouts along arbitrary morphisms of <span class="mathjax-tex">$\mathcal {C}$</span> (in particular these pushouts exist in <span class="mathjax-tex">$\mathcal {C}$</span>). Acyclic cofibrations are stable under pushouts along arbitrary morphisms of <span class="mathjax-tex">$\mathcal {C}$</span>.</p> </li> <li> <p>(C5) Every morphism of <span class="mathjax-tex">$\mathcal {C}$</span> factors as a composite of a cofibration followed by a weak equivalence.</p> </li> <li> <p>(C6) Cofibrations are stable under sequential colimits, i.e. given a sequence of cofibrations </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-b"><figure><div class="c-article-section__figure-content" id="Figb"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figb_HTML.gif?as=webp"><img aria-describedby="Figb" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figb_HTML.gif" alt="figure b" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-b-desc"></div></div></figure></div><p> its colimit <span class="mathjax-tex">$A_\infty $</span> exists and the induced morphism <span class="mathjax-tex">$A_0 \rightarrow A_\infty $</span> is a cofibration. Acyclic cofibrations are stable under sequential colimits.</p> </li> <li> <p>(C7-<span class="mathjax-tex">$\kappa $</span>) Coproducts of <span class="mathjax-tex">$\kappa $</span>-small families of objects exist. Cofibrations and acyclic cofibrations are stable under <span class="mathjax-tex">$\kappa $</span>-small coproducts.</p> </li> </ul> <p>The last two axioms are optional. If we drop them, then cofibration categories can be considered as models of finitely cocomplete homotopy theories. If we include (C6) and (C7-<span class="mathjax-tex">$\kappa $</span>) for a fixed regular cardinal <span class="mathjax-tex">$\kappa > \aleph _0$</span>, we obtain models of <span class="mathjax-tex">$\kappa $</span>-cocomplete homotopy theories, we call them (<i>homotopy</i>) <span class="mathjax-tex">$\kappa $</span> <i>-cocomplete cofibration categories</i>. For <span class="mathjax-tex">$\kappa = \aleph _0$</span> the name (<i>homotopy</i>) <span class="mathjax-tex">$\aleph _0$</span> <i>-cocomplete cofibration category</i> will refer to a cofibration category satisfying the axioms (C0–5). The definition readily dualizes to yield <i>fibration categories</i> which are models of finitely complete homotopy theories or <span class="mathjax-tex">$\kappa $</span>-complete homotopy theories depending on the choice of axioms.</p><p>The main result of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Szumiło, K.: Homotopy theory of cofibration categories. Homol. Homotop. Appl (to appear)" href="/article/10.1007/s40062-016-0139-x#ref-CR21" id="ref-link-section-d81634044e1651">21</a>] establishes the homotopy theory of cofibration categories in the form of a fibration category. We recall the prerequisite definitions before stating the theorem.</p> <h3 class="c-article__sub-heading" id="FPar2">Definition 1.2</h3> <p>A functor <span class="mathjax-tex">$F :\mathcal {C} \rightarrow \mathcal {D}$</span> between cofibration categories is <i>exact</i> if it preserves cofibrations, acyclic cofibrations, initial objects and pushouts along cofibrations.</p> <p>If <span class="mathjax-tex">$\mathcal {C}$</span> and <span class="mathjax-tex">$\mathcal {D}$</span> are <span class="mathjax-tex">$\kappa $</span>-cocomplete, then <i>F</i> is <span class="mathjax-tex">$\kappa $</span> <i>-cocontinuous</i> if, in addition, it preserves colimits of sequences of cofibrations and <span class="mathjax-tex">$\kappa $</span>-small coproducts.</p> <p>The category of (small) <span class="mathjax-tex">$\kappa $</span>-cocomplete cofibration categories and <span class="mathjax-tex">$\kappa $</span>-cocontinuous functors will be denoted by <span class="mathjax-tex">$\mathsf {CofCat}_\kappa $</span>. It is equipped with classes of weak equivalences and fibrations as defined below.</p> <h3 class="c-article__sub-heading" id="FPar3">Definition 1.3</h3> <p>An exact functor <span class="mathjax-tex">$F :\mathcal {C} \rightarrow \mathcal {D}$</span> is a <i>weak equivalence</i> if it induces an equivalence <span class="mathjax-tex">${\text {Ho}}\mathcal {C} \rightarrow {\text {Ho}}\mathcal {D}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar4">Definition 1.4</h3> <p>[<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Szumiło, K.: Homotopy theory of cofibration categories. Homol. Homotop. Appl (to appear)" href="/article/10.1007/s40062-016-0139-x#ref-CR21" id="ref-link-section-d81634044e1962">21</a>, Definition 2.3] Let <span class="mathjax-tex">$P :\mathcal {E} \rightarrow \mathcal {D}$</span> be an exact functor of cofibration categories.</p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(1)</span> <p> <i>P</i> is an <i>isofibration</i> if for every object <span class="mathjax-tex">$A \in \mathcal {E}$</span> and an isomorphism <span class="mathjax-tex">$g :P A \rightarrow Y$</span> there is an isomorphism <span class="mathjax-tex">$f :A \rightarrow B$</span> such that <span class="mathjax-tex">$P f = g$</span>.</p> </li> <li> <span class="u-custom-list-number">(2)</span> <p>It is said to satisfy the <i>lifting property for factorizations</i> if for any morphism <span class="mathjax-tex">$f :A \rightarrow B$</span> of <span class="mathjax-tex">$\mathcal {E}$</span> and a factorization </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-c"><figure><div class="c-article-section__figure-content" id="Figc"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figc_HTML.gif?as=webp"><img aria-describedby="Figc" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figc_HTML.gif" alt="figure c" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-c-desc"></div></div></figure></div><p> there exists a factorization </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-d"><figure><div class="c-article-section__figure-content" id="Figd"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figd_HTML.gif?as=webp"><img aria-describedby="Figd" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figd_HTML.gif" alt="figure d" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-d-desc"></div></div></figure></div><p> such that <span class="mathjax-tex">$Pi = j$</span> and <span class="mathjax-tex">$Ps = t$</span> (in particular, <span class="mathjax-tex">$PC = X$</span>).</p> </li> <li> <span class="u-custom-list-number">(3)</span> <p>It has the <i>lifting property for pseudofactorizations</i> if for any morphism <span class="mathjax-tex">$f :A \rightarrow B$</span> of <span class="mathjax-tex">$\mathcal {E}$</span> and a diagram </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-e"><figure><div class="c-article-section__figure-content" id="Fige"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Fige_HTML.gif?as=webp"><img aria-describedby="Fige" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Fige_HTML.gif" alt="figure e" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-e-desc"></div></div></figure></div><p> there exists a diagram </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-f"><figure><div class="c-article-section__figure-content" id="Figf"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figf_HTML.gif?as=webp"><img aria-describedby="Figf" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figf_HTML.gif" alt="figure f" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-f-desc"></div></div></figure></div><p> such that <span class="mathjax-tex">$Pi = j$</span>, <span class="mathjax-tex">$Ps = t$</span> and <span class="mathjax-tex">$Pu = v$</span> (in particular, <span class="mathjax-tex">$PC = X$</span> and <span class="mathjax-tex">$PD = Y$</span>).</p> </li> <li> <span class="u-custom-list-number">(4)</span> <p>We say that <i>P</i> is a <i>fibration</i> if it is an isofibration and satisfies the lifting properties for factorizations and pseudofactorizations.</p> </li> </ol> <h3 class="c-article__sub-heading" id="FPar5">Theorem 1.5</h3> <p>[<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Szumiło, K.: Homotopy theory of cofibration categories. Homol. Homotop. Appl (to appear)" href="/article/10.1007/s40062-016-0139-x#ref-CR21" id="ref-link-section-d81634044e2552">21</a>, Theorem 2.8] The category <span class="mathjax-tex">$\mathsf {CofCat}_\kappa $</span> of small <span class="mathjax-tex">$\kappa $</span>-cocomplete cofibration categories with weak equivalences and fibrations as above is a fibration category.</p> <p>In the remainder of this section we will introduce a general technique of constructing fibrations of cofibration categories which relies on the notions of direct categories and Reedy cofibrations. We will not discuss the basic theory of Reedy cofibrations since it is already well covered in the literature. A good general reference is [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Riehl, E., Verity, D.: The theory and practice of Reedy categories. Theory Appl. Categ. 29, 256–301 (2014)" href="/article/10.1007/s40062-016-0139-x#ref-CR17" id="ref-link-section-d81634044e2605">17</a>] which is written from the perspective of Reedy categories and model categories. The theory of diagrams over general Reedy categories requires using both colimits and limits. Thus in the case of cofibration categories we have to restrict attention to a special class of Reedy categories called direct categories where colimits suffice. Specific results concerning Reedy cofibrations in cofibration categories are explained in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Răadulescu-Banu, A.: Cofibrations in Homotopy Theory (2006). 
 http://arxiv.org/abs/math/0610009v4
 
 
 " href="/article/10.1007/s40062-016-0139-x#ref-CR14" id="ref-link-section-d81634044e2608">14</a>] from where we will cite a few most relevant to the purpose of this paper.</p> <h3 class="c-article__sub-heading" id="FPar6">Definition 1.6</h3> <ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(1)</span> <p>A category <i>I</i> is <i>direct</i> if it admits a functor <span class="mathjax-tex">$\deg :I \rightarrow \mathbb {N}$</span> that reflects identities, i.e. if <span class="mathjax-tex">$\varphi :i \rightarrow j$</span> is a morphism of <i>I</i> such that <span class="mathjax-tex">$\deg i = \deg j$</span>, then <span class="mathjax-tex">$i = j$</span> and <span class="mathjax-tex">$\varphi = {\mathrm {id}}_i$</span> (we consider <span class="mathjax-tex">$\mathbb {N}$</span> as a poset with its standard order).</p> </li> <li> <span class="u-custom-list-number">(2)</span> <p>For a direct category <i>I</i> and <span class="mathjax-tex">$i \in I$</span>, the <i>latching category</i> at <i>i</i> is the full subcategory of the slice <span class="mathjax-tex">$I {\downarrow }i$</span> on all objects except for <span class="mathjax-tex">${\mathrm {id}}_i$</span>. It is denoted by <span class="mathjax-tex">$\mathord \partial (I {\downarrow }i)$</span>.</p> </li> <li> <span class="u-custom-list-number">(3)</span> <p>Let <span class="mathjax-tex">$X :I \rightarrow \mathcal {C}$</span> be a diagram in some category and <span class="mathjax-tex">$i \in I$</span>. The <i>latching object</i> of <i>X</i> at <i>i</i> is the colimit of the composite diagram </p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \mathord \partial (I {\downarrow }i) \rightarrow I \rightarrow \mathcal {C} \end{aligned}$$</span></div></div><p> where <span class="mathjax-tex">$\mathord \partial (I {\downarrow }i) \rightarrow I$</span> is the forgetful functor sending a morphism of <i>I</i> (i.e. an object of <span class="mathjax-tex">$\mathord \partial (I {\downarrow }i)$</span>) to its source. The latching object (if it exists) is denoted by <span class="mathjax-tex">$L_i X$</span> and comes with a canonical <i>latching morphism</i> <span class="mathjax-tex">$L_i X \rightarrow X_i$</span> induced by the inclusion <span class="mathjax-tex">$\mathord \partial (I {\downarrow }i) \rightarrow I {\downarrow }i$</span>.</p> </li> <li> <span class="u-custom-list-number">(4)</span> <p>Let <span class="mathjax-tex">$\mathcal {C}$</span> be a cofibration category. A diagram <span class="mathjax-tex">$X :I \rightarrow \mathcal {C}$</span> is <i>Reedy cofibrant</i> if for all <span class="mathjax-tex">$i \in I$</span> the latching object of <i>X</i> at <i>i</i> exists and the latching morphism <span class="mathjax-tex">$L_i X \rightarrow X_i$</span> is a cofibration.</p> </li> <li> <span class="u-custom-list-number">(5)</span> <p>Let <span class="mathjax-tex">$f :X \rightarrow Y$</span> be a morphism of Reedy cofibrant diagrams <span class="mathjax-tex">$I \rightarrow \mathcal {C}$</span>. It is called a <i>Reedy cofibration</i> if for all <span class="mathjax-tex">$i \in I$</span> the induced morphism </p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} X_i \amalg _{L_i X} L_i Y \rightarrow Y_i \end{aligned}$$</span></div></div><p> is a cofibration (observe that this pushout exists since <i>X</i> is Reedy cofibrant).</p> </li> </ol> <p>The main purpose of this section is to construct certain cofibration categories of diagrams and establish some practical criteria for verifying that particular functors between them are weak equivalences or fibrations.</p> <h3 class="c-article__sub-heading" id="FPar7">Proposition 1.7</h3> <p>Let <span class="mathjax-tex">$\mathcal {C}$</span> be a cofibration category and <i>J</i> a homotopical direct category with finite latching categories.</p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(1)</span> <p>The category <span class="mathjax-tex">$\mathcal {C}^J_\mathrm {R}$</span> of homotopical Reedy cofibrant diagrams with levelwise weak equivalences and Reedy cofibrations is a cofibration category.</p> </li> <li> <span class="u-custom-list-number">(2)</span> <p>The category <span class="mathjax-tex">$\mathcal {C}^J$</span> of all homotopical diagrams with levelwise weak equivalences and levelwise cofibrations is a cofibration category.</p> </li> <li> <span class="u-custom-list-number">(3)</span> <p>The inclusion functor <span class="mathjax-tex">$\mathcal {C}^J_\mathrm {R}\hookrightarrow \mathcal {C}^J$</span> is a weak equivalence.</p> </li> </ol> <h3 class="c-article__sub-heading" id="FPar8">Proof</h3> <ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(1)</span> <p>[<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Răadulescu-Banu, A.: Cofibrations in Homotopy Theory (2006). 
 http://arxiv.org/abs/math/0610009v4
 
 
 " href="/article/10.1007/s40062-016-0139-x#ref-CR14" id="ref-link-section-d81634044e3804">14</a>, Theorem 9.3.8 (1a)]</p> </li> <li> <span class="u-custom-list-number">(2)</span> <p>[<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Răadulescu-Banu, A.: Cofibrations in Homotopy Theory (2006). 
 http://arxiv.org/abs/math/0610009v4
 
 
 " href="/article/10.1007/s40062-016-0139-x#ref-CR14" id="ref-link-section-d81634044e3818">14</a>, Theorem 9.3.8 (1b)]</p> </li> <li> <span class="u-custom-list-number">(3)</span> <p>The inclusion functor satisfies the approximation properties of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Szumiło, K.: Homotopy theory of cofibration categories. Homol. Homotop. Appl (to appear)" href="/article/10.1007/s40062-016-0139-x#ref-CR21" id="ref-link-section-d81634044e3832">21</a>, Proposition 2.2] as follows from Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar11">1.9</a> (1) (in fact, from its standard special case of <span class="mathjax-tex">$\mathcal {D} = [0]$</span> and <span class="mathjax-tex">$I = \varnothing $</span>). <span class="mathjax-tex">$\square $</span> </p> </li> </ol> <p>The crucial step in the proof of the above proposition is the construction of factorizations. In Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar11">1.9</a> we revisit that construction in order to prove a more general version which will be a key technical tool in many arguments of this paper.</p><p>A homotopical functor <span class="mathjax-tex">$f :I \rightarrow J$</span> is a <i>homotopy equivalence</i> if there is a homotopical functor <span class="mathjax-tex">$g :J \rightarrow I$</span> such that <i>gf</i> is weakly equivalent to <span class="mathjax-tex">${\mathrm {id}}_I$</span> and <i>fg</i> is weakly equivalent to <span class="mathjax-tex">${\mathrm {id}}_J$</span> (where “weakly equivalent” means “connected by a zig-zag of natural weak equivalences”).</p> <h3 class="c-article__sub-heading" id="FPar9">Lemma 1.8</h3> <p>Let <span class="mathjax-tex">$\mathcal {C}$</span> be a cofibration category and <span class="mathjax-tex">$f :I \rightarrow J$</span> a homotopical functor where <i>I</i> and <i>J</i> are homotopical direct categories with finite latching categories.</p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(1)</span> <p>The induced functor <span class="mathjax-tex">$f^* :\mathcal {C}^J \rightarrow \mathcal {C}^I$</span> is exact.</p> </li> <li> <span class="u-custom-list-number">(2)</span> <p>If <i>f</i> is a homotopy equivalence, then <span class="mathjax-tex">$f^* :\mathcal {C}^J \rightarrow \mathcal {C}^I$</span> is a weak equivalence of cofibration categories.</p> </li> <li> <span class="u-custom-list-number">(3)</span> <p>If <i>f</i> is a homotopy equivalence and induces an exact functor <span class="mathjax-tex">$f^* :\mathcal {C}^J_\mathrm {R}\rightarrow \mathcal {C}^I_\mathrm {R}$</span>, then it is also a weak equivalence.</p> </li> </ol> <h3 class="c-article__sub-heading" id="FPar10">Proof</h3> <p>The functor <span class="mathjax-tex">$f^*$</span> is clearly exact with respect to the levelwise structures and it is a homotopy equivalence when <i>f</i> is.</p> <p>For the last statement, consider the commutative square of exact functors </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-g"><figure><div class="c-article-section__figure-content" id="Figg"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figg_HTML.gif?as=webp"><img aria-describedby="Figg" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figg_HTML.gif" alt="figure g" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-g-desc"></div></div></figure></div> <p>the vertical maps are weak equivalences by Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar7">1.7</a> so the conclusion follows by 2-out-of-3. <span class="mathjax-tex">$\square $</span> </p> <p>The utility of direct categories comes from the fact that it is easy to construct diagrams and morphisms of diagrams inductively. For our purposes it will be most convenient to state this in terms of sieves. A functor <span class="mathjax-tex">$I \rightarrow J$</span> is called a <i>sieve</i> if it is an inclusion of a full downwards closed subcategory, i.e. if it is injective on objects, fully faithful and if <span class="mathjax-tex">$i \rightarrow j$</span> is a morphism of <i>J</i> such that <span class="mathjax-tex">$j \in I$</span>, then <span class="mathjax-tex">$i \in I$</span>. If <i>I</i> and <i>J</i> are homotopical categories, we will further assume as a part of the definition of a sieve <span class="mathjax-tex">$I \rightarrow J$</span> that it preserves and reflects weak equivalences, i.e. a morphism of <i>I</i> is a weak equivalence if and only if its image in <i>J</i> is.</p><p>The first part of the next lemma generalizes the standard construction of factorizations into Reedy cofibrations followed by weak equivalences. It says that given a morphism of diagrams <span class="mathjax-tex">$J \rightarrow \mathcal {C}$</span> and compatible factorizations of its restriction along a sieve <span class="mathjax-tex">$I \hookrightarrow J$</span> and its image under a fibration <span class="mathjax-tex">$P :\mathcal {C} \rightarrow \mathcal {D}$</span>, there is a factorization of the original morphism compatible with both of them. The other two parts say the same for lifts for pseudofactorizations and for cofibrations (when <i>P</i> is an acyclic fibration as in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Szumiło, K.: Homotopy theory of cofibration categories. Homol. Homotop. Appl (to appear)" href="/article/10.1007/s40062-016-0139-x#ref-CR21" id="ref-link-section-d81634044e4648">21</a>, Proposition 2.5]).</p> <h3 class="c-article__sub-heading" id="FPar11">Lemma 1.9</h3> <p>Let <span class="mathjax-tex">$P :\mathcal {C} \twoheadrightarrow \mathcal {D}$</span> be a fibration between cofibration categories. Let <i>J</i> be a homotopical direct category with finite latching categories and <span class="mathjax-tex">$I \hookrightarrow J$</span> a sieve.</p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(1)</span> <p>Let <span class="mathjax-tex">$f :X \rightarrow Y$</span> be a morphism in <span class="mathjax-tex">$\mathcal {C}^J$</span>. If <i>X</i> is Reedy cofibrant, </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-h"><figure><div class="c-article-section__figure-content" id="Figh"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figh_HTML.gif?as=webp"><img aria-describedby="Figh" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figh_HTML.gif" alt="figure h" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-h-desc"></div></div></figure></div><p> are factorizations of <i>Pf</i> and <i>f</i>|<i>I</i> into Reedy cofibrations followed by weak equivalences such that <span class="mathjax-tex">$Pk_I = k_P|I$</span> and <span class="mathjax-tex">$Ps_I = s_P|I$</span> (in particular, <span class="mathjax-tex">$P\widetilde{Y}_I = \widetilde{Y}_P|I$</span>), then there is a factorization </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-i"><figure><div class="c-article-section__figure-content" id="Figi"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figi_HTML.gif?as=webp"><img aria-describedby="Figi" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figi_HTML.gif" alt="figure i" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-i-desc"></div></div></figure></div><p> of <i>f</i> into a Reedy cofibration followed by a weak equivalence such that <span class="mathjax-tex">$Pk = k_P$</span>, <span class="mathjax-tex">$k|I = k_I$</span>, <span class="mathjax-tex">$Ps = s_P$</span> and <span class="mathjax-tex">$s|I = s_I$</span> (in particular, <span class="mathjax-tex">$P\widetilde{Y} = \widetilde{Y}_P$</span> and <span class="mathjax-tex">$\widetilde{Y}|I = \widetilde{Y}_I$</span>).</p> </li> <li> <span class="u-custom-list-number">(2)</span> <p>Let <span class="mathjax-tex">$f :X \rightarrow Y$</span> be a morphism in <span class="mathjax-tex">$\mathcal {C}^J$</span>. If both <i>X</i> and <i>Y</i> are Reedy cofibrant, </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-j"><figure><div class="c-article-section__figure-content" id="Figj"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figj_HTML.gif?as=webp"><img aria-describedby="Figj" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figj_HTML.gif" alt="figure j" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-j-desc"></div></div></figure></div><p> are pseudofactorizations of <i>Pf</i> and <i>f</i>|<i>I</i> such that <span class="mathjax-tex">$Pk = k_p$</span>, <span class="mathjax-tex">$k|I = k_I$</span>, <span class="mathjax-tex">$Pl = l_P$</span>, <span class="mathjax-tex">$l|I = l_I$</span>, <span class="mathjax-tex">$Ps = s_P$</span> and <span class="mathjax-tex">$s|I = s_I$</span> (in particular, <span class="mathjax-tex">$P\widetilde{Y}_I = \widetilde{Y}_P|I$</span> and <span class="mathjax-tex">$P\widehat{Y}_I = \widehat{Y}_P|I$</span>), then there is a pseudofactorization </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-k"><figure><div class="c-article-section__figure-content" id="Figk"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figk_HTML.gif?as=webp"><img aria-describedby="Figk" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figk_HTML.gif" alt="figure k" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-k-desc"></div></div></figure></div><p> such that <span class="mathjax-tex">$Pk = k_P$</span>, <span class="mathjax-tex">$k|I = k_I$</span>, <span class="mathjax-tex">$Pl = l_P$</span>, <span class="mathjax-tex">$l|I = l_I$</span>, <span class="mathjax-tex">$Ps = s_P$</span> and <span class="mathjax-tex">$s|I = s_I$</span> (in particular, <span class="mathjax-tex">$P\widetilde{Y} = \widetilde{Y}_P$</span>, <span class="mathjax-tex">$\widetilde{Y}|I = \widetilde{Y}_I$</span>, <span class="mathjax-tex">$P\widehat{Y} = \hat{Y}_P$</span> and <span class="mathjax-tex">$\widehat{Y}|I = \widehat{Y}_I$</span>).</p> </li> <li> <span class="u-custom-list-number">(3)</span> <p>If <i>P</i> is acyclic, <i>X</i> is a Reedy cofibrant diagram in <span class="mathjax-tex">$\mathcal {C}^J$</span> and </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-l"><figure><div class="c-article-section__figure-content" id="Figl"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figl_HTML.gif?as=webp"><img aria-describedby="Figl" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figl_HTML.gif" alt="figure l" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-l-desc"></div></div></figure></div><p> are Reedy cofibrations such that <span class="mathjax-tex">$P k_I = k_P|I$</span>, then there exists a Reedy cofibration </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-m"><figure><div class="c-article-section__figure-content" id="Figm"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figm_HTML.gif?as=webp"><img aria-describedby="Figm" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figm_HTML.gif" alt="figure m" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-m-desc"></div></div></figure></div><p> such that <span class="mathjax-tex">$Pk = k_P$</span> and <span class="mathjax-tex">$k|I = k_I$</span> (in particular, <span class="mathjax-tex">$PZ = Z_P$</span> and <span class="mathjax-tex">$Z|I = Z_I$</span>).</p> </li> </ol> <h3 class="c-article__sub-heading" id="FPar12">Proof</h3> <p>The proofs of three parts are similar to each other so we only provide the first one. (the second one uses the lifting property for pseudofactorizations and the third one uses the lifting property of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Szumiło, K.: Homotopy theory of cofibration categories. Homol. Homotop. Appl (to appear)" href="/article/10.1007/s40062-016-0139-x#ref-CR21" id="ref-link-section-d81634044e6322">21</a>, Proposition 2.5]).</p> <p>It suffices to extend the factorization <span class="mathjax-tex">$f|I = s_I k_I$</span> over an object <span class="mathjax-tex">$j \in J{\setminus }I$</span> of a minimal degree. Then the statement will follow by an induction over the degree.</p> <p>By the minimality of the degree of <i>j</i>, Reedy cofibrancy of <i>X</i> and since <span class="mathjax-tex">$I \hookrightarrow J$</span> is a sieve the latching objects <span class="mathjax-tex">$L_j X$</span> and <span class="mathjax-tex">$L_j \widetilde{Y}_I$</span> exist. Moreover, the induced functor of latching categories <span class="mathjax-tex">$\mathord \partial (I {\downarrow }j) \rightarrow \mathord \partial (J {\downarrow }j)$</span> is an isomorphism. Thus <i>P</i> sends the morphism <span class="mathjax-tex">$X_j \amalg _{L_j X} L_j \widetilde{Y}_I \rightarrow Y_j$</span> to the analogous morphism <span class="mathjax-tex">$PX_j \amalg _{L_j PX} P\widetilde{Y}_I \rightarrow P Y_j$</span>. The latter factors as</p><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} PX_j \amalg _{L_j PX} P\widetilde{Y}_I \rightarrowtail (\widetilde{Y}_P)_j \mathop {\rightarrow }\limits ^{\sim }P Y_j \end{aligned}$$</span></div></div><p>and since <i>P</i> is a fibration we can lift this to a factorization of the former as</p><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} X_j \amalg _{L_j X} L_j \widetilde{Y}_I \rightarrowtail \widetilde{Y}_j \mathop {\rightarrow }\limits ^{\sim }Y_j \text {.} \end{aligned}$$</span></div></div><p>This extends the factorization <span class="mathjax-tex">$f|I = s_I k_I$</span> over <i>j</i>.</p> <p>The resulting diagram <span class="mathjax-tex">$\widetilde{Y}$</span> is homotopical since it is weakly equivalent to homotopical <i>Y</i>. <span class="mathjax-tex">$\square $</span> </p> <p>The most typical examples of fibrations are restrictions along sieves.</p> <h3 class="c-article__sub-heading" id="FPar13">Lemma 1.10</h3> <p>Let <span class="mathjax-tex">$\mathcal {C}$</span> be a cofibration category. If <i>I</i> and <i>J</i> are homotopical direct categories with finite latching categories and <span class="mathjax-tex">$f :I \rightarrow J$</span> a homotopical functor such that for every <span class="mathjax-tex">$i \in I$</span> the induced functor of the latching categories <span class="mathjax-tex">$\mathord \partial (I {\downarrow }i) \rightarrow \mathord \partial (J {\downarrow }fi)$</span> is an isomorphism, then the functor <span class="mathjax-tex">$f^* :\mathcal {C}^J \rightarrow \mathcal {C}^I$</span> induces a functor <span class="mathjax-tex">$f^* :\mathcal {C}^J_\mathrm {R}\rightarrow \mathcal {C}^I_\mathrm {R}$</span> which is exact.</p> <p>Moreover, if <i>f</i> is a sieve, then <span class="mathjax-tex">$f^*$</span> is a fibration.</p> <h3 class="c-article__sub-heading" id="FPar14">Proof</h3> <p>If <i>f</i> induces isomorphisms of the latching categories, then <span class="mathjax-tex">$f^*$</span> preserves Reedy cofibrations (and, in particular, Reedy cofibrant diagrams). It also preserves weak equivalences and colimits that exist in <span class="mathjax-tex">$\mathcal {C}^J_\mathrm {R}$</span> so it is exact.</p> <p>If <i>f</i> is a sieve, then it satisfies the exactness criterion above. Moreover, <span class="mathjax-tex">$f^*$</span> is a fibration by parts (1) and (2) of Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar11">1.9</a>. <span class="mathjax-tex">$\square $</span> </p> <p>The next few lemmas establish some connections between sieves and fibrations which are reminiscent of classical homotopical algebra if we think of sieves as “cofibrations” and sieves <span class="mathjax-tex">$I \hookrightarrow J$</span> inducing weak equivalences <span class="mathjax-tex">$\mathcal {C}^J_\mathrm {R}\rightarrow \mathcal {C}^I_\mathrm {R}$</span> as “acyclic cofibrations”. This does not quite fit into the classical picture since such “cofibrations” do not really belong to the same category as the fibrations. The situation bears some resemblance to the “pushout product property” of a simplicial model category <span class="mathjax-tex">$\mathcal {M}$</span> (see e.g. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Hovey, M.: Model categories, Mathematical Surveys and Monographs, vol. 63. American Mathematical Society, Providence, RI (1999)" href="/article/10.1007/s40062-016-0139-x#ref-CR8" id="ref-link-section-d81634044e7575">8</a>, Definition 4.2.18]), but is different. In the present context it is essential that Reedy cofibrant diagrams can be seen as “morphisms” from direct categories to cofibration categories, while in the context of the pushout product property there is usually no meaningful notion of a morphism from a simplicial set to an object of <span class="mathjax-tex">$\mathcal {M}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar15">Lemma 1.11</h3> <p>Let <span class="mathjax-tex">$f :I \hookrightarrow J$</span> be a sieve between homotopical direct categories with finite latching categories and <span class="mathjax-tex">$P :\mathcal {C} \rightarrow \mathcal {D}$</span> a fibration of cofibration categories. Then the induced exact functor <span class="mathjax-tex">$(f^*, P) :\mathcal {C}^J_\mathrm {R}\rightarrow \mathcal {C}^I_\mathrm {R}\times _{\mathcal {D}^I_\mathrm {R}} \mathcal {D}^J_\mathrm {R}$</span> </p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(1)</span> <p>is a fibration,</p> </li> <li> <span class="u-custom-list-number">(2)</span> <p>is an acyclic fibration provided that <i>P</i> is acyclic,</p> </li> <li> <span class="u-custom-list-number">(3)</span> <p>is an acyclic fibration provided that both <span class="mathjax-tex">$f^* :\mathcal {C}^J_\mathrm {R}\rightarrow \mathcal {C}^I_\mathrm {R}$</span> and <span class="mathjax-tex">$f^* :\mathcal {D}^J_\mathrm {R}\rightarrow \mathcal {D}^I_\mathrm {R}$</span> are weak equivalences.</p> </li> </ol> <h3 class="c-article__sub-heading" id="FPar16">Proof</h3> <p>First observe that the pullback in question exists since <span class="mathjax-tex">$f^*$</span> is a fibration by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar13">1.10</a>.</p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(1)</span> <p>This follows by parts (1) and (2) of Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar11">1.9</a>.</p> </li> <li> <span class="u-custom-list-number">(2)</span> <p>This follows by (1) above and part (3) of Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar11">1.9</a>.</p> </li> <li> <span class="u-custom-list-number">(2)</span> <p>This follows by (1) above and a diagram chase using the fact that acyclic fibrations are closed under pullbacks.</p> </li> </ol><p> <span class="mathjax-tex">$\square $</span> </p> <h3 class="c-article__sub-heading" id="FPar17">Lemma 1.12</h3> <p>If <span class="mathjax-tex">$\mathcal {C}$</span> is a cofibration category, </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-n"><figure><div class="c-article-section__figure-content" id="Fign"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Fign_HTML.gif?as=webp"><img aria-describedby="Fign" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Fign_HTML.gif" alt="figure n" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-n-desc"></div></div></figure></div> <p>is a pushout square of homotopical direct categories with finite latching categories and both <span class="mathjax-tex">$I \hookrightarrow J$</span> and <span class="mathjax-tex">$I \hookrightarrow K$</span> are sieves, then the resulting square </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-o"><figure><div class="c-article-section__figure-content" id="Figo"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figo_HTML.gif?as=webp"><img aria-describedby="Figo" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figo_HTML.gif" alt="figure o" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-o-desc"></div></div></figure></div> <p>is a pullback of cofibration categories.</p> <h3 class="c-article__sub-heading" id="FPar18">Proof</h3> <p>By the construction of pullbacks of cofibration categories it will suffice to verify that a morphism of diagrams over <i>L</i> is a Reedy cofibration if and only if it is one when restricted to both <i>J</i> and <i>K</i>. For this it will be enough to observe that both <span class="mathjax-tex">$J \hookrightarrow L$</span> and <span class="mathjax-tex">$K \hookrightarrow L$</span> are sieves and hence for an object <span class="mathjax-tex">$l \in L$</span> we have either <span class="mathjax-tex">$l \in J$</span> and then <span class="mathjax-tex">$\mathord \partial (J {\downarrow }l) \rightarrow \mathord \partial (L {\downarrow }l)$</span> is an isomorphism or <span class="mathjax-tex">$l \in K$</span> and then <span class="mathjax-tex">$\mathord \partial (K {\downarrow }l) \rightarrow \mathord \partial (L {\downarrow }l)$</span> is an isomorphism. <span class="mathjax-tex">$\square $</span> </p> <p>Let <span class="mathjax-tex">$f :I \rightarrow J$</span> be a homotopical functor of homotopical direct categories and <span class="mathjax-tex">$F :\mathcal {C} \rightarrow \mathcal {D}$</span> an exact functor of cofibration categories. We say that <i>f</i> has the Reedy left lifting property with respect to <i>F</i> (or <i>F</i> has the Reedy right lifting property with respect to <i>f</i>) if every lifting problem </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-p"><figure><div class="c-article-section__figure-content" id="Figp"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figp_HTML.gif?as=webp"><img aria-describedby="Figp" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figp_HTML.gif" alt="figure p" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-p-desc"></div></div></figure></div> <p>where <i>X</i> and <i>Y</i> are homotopical Reedy cofibrant diagrams has a solution that is also a homotopical Reedy cofibrant diagram.</p> <h3 class="c-article__sub-heading" id="FPar19">Lemma 1.13</h3> <p>Let <span class="mathjax-tex">$f :I \hookrightarrow J$</span> and <span class="mathjax-tex">$g :K \rightarrow L$</span> be sieves between homotopical direct categories with finite latching categories and <span class="mathjax-tex">$F :\mathcal {C} \rightarrow \mathcal {D}$</span> an exact functor of cofibration categories. Then there is a natural bijection between Reedy lifting problems (and their solutions) of the forms </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-q"><figure><div class="c-article-section__figure-content" id="Figq"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figq_HTML.gif?as=webp"><img aria-describedby="Figq" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figq_HTML.gif" alt="figure q" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-q-desc"></div></div></figure></div> <h3 class="c-article__sub-heading" id="FPar20">Proof</h3> <p>This this will follow from standard adjointness arguments, e.g. as in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Joyal, A.: The Theory of Quasi-Categories and its Applications. Quadern 45, Vol. II, Centre de Recerca Matemàtica Barcelona (2008)" href="/article/10.1007/s40062-016-0139-x#ref-CR10" id="ref-link-section-d81634044e8635">10</a>, Proposition D.1.18], if we can verify that a diagram <span class="mathjax-tex">$X :J \times L \rightarrow \mathcal {C}$</span> is Reedy cofibrant if and only if the corresponding diagram <span class="mathjax-tex">$\widetilde{X} :J \rightarrow \mathcal {C}^L$</span> is Reedy cofibrant as a diagram <span class="mathjax-tex">$J \rightarrow \mathcal {C}^L_\mathrm {R}$</span>.</p> <p>First, assume that <i>X</i> is Reedy cofibrant. For <span class="mathjax-tex">$j \in J$</span>, the diagrams <span class="mathjax-tex">$\widetilde{X}_j, L_j \widetilde{X} :L \rightarrow \mathcal {C}$</span> can be computed as the left Kan extensions of (restrictions of) <i>X</i> along the projections <span class="mathjax-tex">$(J {\downarrow }j) \times L \rightarrow L$</span> and <span class="mathjax-tex">$\mathord \partial (J {\downarrow }j) \times L \rightarrow L$</span>. These Kan extensions exist and are Reedy cofibrant by [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Răadulescu-Banu, A.: Cofibrations in Homotopy Theory (2006). 
 http://arxiv.org/abs/math/0610009v4
 
 
 " href="/article/10.1007/s40062-016-0139-x#ref-CR14" id="ref-link-section-d81634044e8946">14</a>, Theorem 9.4.3 (1)]. Moreover, it follows from [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Răadulescu-Banu, A.: Cofibrations in Homotopy Theory (2006). 
 http://arxiv.org/abs/math/0610009v4
 
 
 " href="/article/10.1007/s40062-016-0139-x#ref-CR14" id="ref-link-section-d81634044e8949">14</a>, Theorem 9.4.1 (1)] that the induced morphism <span class="mathjax-tex">$L_j \widetilde{X} \rightarrow \widetilde{X}_j$</span> is a Reedy cofibration since the inclusion <span class="mathjax-tex">$\mathord \partial (J {\downarrow }j) \times L \hookrightarrow (J {\downarrow }j) \times L$</span> is a sieve. Thus <span class="mathjax-tex">$\widetilde{X}$</span> is a Reedy cofibrant diagram <span class="mathjax-tex">$J \rightarrow \mathcal {C}^L_\mathrm {R}$</span>.</p> <p>Conversely, assume that <span class="mathjax-tex">$\widetilde{X}$</span> is a Reedy cofibrant diagram <span class="mathjax-tex">$J \rightarrow \mathcal {C}^L_\mathrm {R}$</span>. For all <span class="mathjax-tex">$j \in J$</span> and <span class="mathjax-tex">$l \in L$</span>, we need to verify that the latching object <span class="mathjax-tex">$L_{j,l} X$</span> exists and the latching morphism <span class="mathjax-tex">$L_{j,l} X \rightarrow X_{j,l}$</span> is a cofibration. Proceeding by induction, we may assume that this true for all objects of <span class="mathjax-tex">$J \times L$</span> lying strictly below (<i>j</i>, <i>l</i>). This implies that the composite diagram</p><div id="Equ5" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \mathord \partial (J \times L {\downarrow }(j,l) ) \rightarrow J \times L \rightarrow \mathcal {C} \end{aligned}$$</span></div></div><p>is Reedy cofibrant and hence <span class="mathjax-tex">$L_{j,l} X$</span> exists. Moreover, there is a pushout square of direct categories </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-r"><figure><div class="c-article-section__figure-content" id="Figr"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figr_HTML.gif?as=webp"><img aria-describedby="Figr" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figr_HTML.gif" alt="figure r" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-r-desc"></div></div></figure></div> <p>(where all maps are sieves) which implies that <span class="mathjax-tex">$L_{j,l}$</span> can also be computed as the pushout </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-s"><figure><div class="c-article-section__figure-content" id="Figs"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figs_HTML.gif?as=webp"><img aria-describedby="Figs" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figs_HTML.gif" alt="figure s" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-s-desc"></div></div></figure></div> <p>i.e. <span class="mathjax-tex">$L_{j,l} X$</span> coincides with the relative latching object of the morphism <span class="mathjax-tex">$L_j \widetilde{X} \rightarrow \widetilde{X}_j$</span> at <i>l</i>. Hence <span class="mathjax-tex">$L_{j,l} X \rightarrow X_{j,l}$</span> is a cofibration since <span class="mathjax-tex">$\widetilde{X}$</span> is Reedy cofibrant. <span class="mathjax-tex">$\square $</span> </p> <h3 class="c-article__sub-heading" id="FPar21">Lemma 1.14</h3> <p>Let <span class="mathjax-tex">$P :\mathcal {C} \rightarrow \mathcal {D}$</span> be a fibration of cofibration categories. The following are equivalent:</p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(1)</span> <p> <i>P</i> is acyclic,</p> </li> <li> <span class="u-custom-list-number">(2)</span> <p> <i>P</i> has the Reedy right lifting property with respect to all sieves between direct homotopical categories with finite latching categories,</p> </li> <li> <span class="u-custom-list-number">(3)</span> <p> <i>P</i> has the Reedy right lifting property with respect to <span class="mathjax-tex">$[0] \hookrightarrow [1]$</span> and <span class="mathjax-tex">$[1] \hookrightarrow \widehat{[1]}$</span>.</p> </li> </ol> <h3 class="c-article__sub-heading" id="FPar22">Proof</h3> <p>If <i>P</i> is acyclic, then it has the Reedy right lifting property with respect to all sieves between homotopical direct categories with finite latching categories by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar11">1.9</a> (3), in particular, with respect to <span class="mathjax-tex">$[0] \hookrightarrow [1]$</span> and <span class="mathjax-tex">$[1] \hookrightarrow \widehat{[1]}$</span>.</p> <p>Conversely, by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar15">1.11</a> it suffices to see that if <i>P</i> has the Reedy right lifting property with respect to <span class="mathjax-tex">$[0] \hookrightarrow [1]$</span> and <span class="mathjax-tex">$[1] \hookrightarrow \widehat{[1]}$</span>, then it satisfies (App1) and has the right lifting property in <span class="mathjax-tex">$\overline{\mathsf {CofCat}}$</span> with respect to the inclusion of [0] into </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-t"><figure><div class="c-article-section__figure-content" id="Figt"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figt_HTML.gif?as=webp"><img aria-describedby="Figt" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figt_HTML.gif" alt="figure t" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-t-desc"></div></div></figure></div><p> The latter is equivalent to the Reedy right lifting property with respect to <span class="mathjax-tex">$[0] \hookrightarrow [1]$</span>. To see that the Reedy right lifting property with respect to <span class="mathjax-tex">$[1] \hookrightarrow \widehat{[1]}$</span> implies (App1) take a morphism <span class="mathjax-tex">$f :X \rightarrow Y$</span> in <span class="mathjax-tex">$\mathcal {C}$</span> such that <i>Pf</i> is a weak equivalence. Factor <i>f</i> as </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-u"><figure><div class="c-article-section__figure-content" id="Figu"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figu_HTML.gif?as=webp"><img aria-describedby="Figu" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figu_HTML.gif" alt="figure u" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-u-desc"></div></div></figure></div><p> Then <i>Pj</i> is a weak equivalence by 2-out-of-3 and hence so is <i>j</i> by the Reedy right lifting property with respect to <span class="mathjax-tex">$[1] \hookrightarrow \widehat{[1]}$</span>. Thus <i>f</i> is a weak equivalence, too. <span class="mathjax-tex">$\square $</span> </p> <h3 class="c-article__sub-heading" id="FPar23">Lemma 1.15</h3> <p>If a sieve <span class="mathjax-tex">$f :I \rightarrow J$</span> between homotopical direct categories has the Reedy left lifting property with respect to all fibrations of cofibration categories, then for every cofibration category <span class="mathjax-tex">$\mathcal {C}$</span> the induced functor <span class="mathjax-tex">$f^* :\mathcal {C}^J_\mathrm {R}\rightarrow \mathcal {C}^I_\mathrm {R}$</span> is an acyclic fibration.</p> <h3 class="c-article__sub-heading" id="FPar24">Proof</h3> <p>Since <i>f</i> is a sieve, <span class="mathjax-tex">$f^*$</span> is a fibration by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar13">1.10</a>. Thus, by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar21">1.14</a>, it will suffice to check that <span class="mathjax-tex">$f^*$</span> has the Reedy right lifting property with respect to <span class="mathjax-tex">$[0] \hookrightarrow [1]$</span> and <span class="mathjax-tex">$[1] \hookrightarrow \widehat{[1]}$</span>. These are equivalent to the Reedy right lifting property of <span class="mathjax-tex">$\mathcal {C}^{[1]}_\mathrm {R}\rightarrow \mathcal {C}^{[0]}_\mathrm {R}$</span> and <span class="mathjax-tex">$\mathcal {C}^{\widehat{[1]}}_\mathrm {R}\rightarrow \mathcal {C}^{[1]}_\mathrm {R}$</span> with respect to <span class="mathjax-tex">$I \hookrightarrow J$</span> by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar19">1.13</a>. <span class="mathjax-tex">$\square $</span> </p> </div></div></section><section data-title="Quasicategories of frames"><div class="c-article-section" id="Sec3-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec3"><span class="c-article-section__title-number">3 </span>Quasicategories of frames</h2><div class="c-article-section__content" id="Sec3-content"><p>Before introducing quasicategories of frames we need to explain a preliminary construction which will play an essential role in the remainder of this paper. It depends on properties of direct and homotopical categories discussed in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s40062-016-0139-x#Sec2">1</a>.</p><p>Let <span class="mathjax-tex">$\Delta _\sharp $</span> denote the subcategory of injective maps in <span class="mathjax-tex">$\Delta $</span> and let <i>J</i> be a homotopical category. We construct a direct homotopical category <i>DJ</i> and a homotopical functor <span class="mathjax-tex">$p_J :DJ \rightarrow J$</span> as follows. The underlying category of <i>DJ</i> is the comma category <span class="mathjax-tex">$\Delta _\sharp {\downarrow }J$</span>, i.e. objects are all functors <span class="mathjax-tex">$[m] \rightarrow J$</span> for all <i>m</i> and a morphism from <span class="mathjax-tex">$x :[m] \rightarrow J$</span> to <span class="mathjax-tex">$y :[n] \rightarrow J$</span> is an injective order preserving map <span class="mathjax-tex">$\varphi :[m] \hookrightarrow [n]$</span> such that <span class="mathjax-tex">$x = y \varphi $</span>. The functor <span class="mathjax-tex">$p_J :\Delta _\sharp {\downarrow }J \rightarrow J$</span> (sometimes called the <i>last vertex projection</i>) is defined by sending <span class="mathjax-tex">$x :[m] \rightarrow J$</span> to <span class="mathjax-tex">$x_m$</span> and a morphism <span class="mathjax-tex">$\varphi $</span> as above to the induced morphism <span class="mathjax-tex">$x_m = y_{\varphi (m)} \rightarrow y_n$</span>. The weak equivalences in <i>DJ</i> are created by <span class="mathjax-tex">$p_J$</span>. Then <i>DJ</i> is homotopical category, <span class="mathjax-tex">$p_J$</span> is a homotopical functor and <i>DJ</i> is also direct (by setting the degree of <span class="mathjax-tex">$[m] \rightarrow J$</span> to <i>m</i>). We can think of <i>DJ</i> as a <i>direct approximation</i> to <i>J</i>. Observe that <i>D</i> is a functor from homotopical categories to homotopical categories and that <i>DJ</i> has a non-trivial homotopical structure even if <i>J</i> has the trivial one (unless <i>J</i> is empty). This construction has multiple motivations which will be given right after the definition of quasicategories of frames below.</p><p>First, we need to verify that Reedy cofibrant diagrams over <i>DJ</i> are well behaved with respect to homotopical functors <span class="mathjax-tex">$I \rightarrow J$</span>. If <i>f</i> is such a functor we will abbreviate the induced functor <span class="mathjax-tex">$(Df)^* :\mathcal {C}^{DJ}_\mathrm {R}\rightarrow \mathcal {C}^{DI}_\mathrm {R}$</span> to <span class="mathjax-tex">$f^*$</span> to simplify the notation. Recall that <span class="mathjax-tex">$\mathcal {C}^{DJ}_\mathrm {R}$</span> refers to the cofibration category of homotopical Reedy cofibrant diagrams <span class="mathjax-tex">$DJ \rightarrow \mathcal {C}$</span> with levelwise weak equivalences and Reedy cofibrations which exists by [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Răadulescu-Banu, A.: Cofibrations in Homotopy Theory (2006). 
 http://arxiv.org/abs/math/0610009v4
 
 
 " href="/article/10.1007/s40062-016-0139-x#ref-CR14" id="ref-link-section-d81634044e11699">14</a>, Theorem 9.3.8 (1a)].</p> <h3 class="c-article__sub-heading" id="FPar25">Lemma 2.1</h3> <p>Let <span class="mathjax-tex">$\mathcal {C}$</span> be a cofibration category. If <span class="mathjax-tex">$f :I \rightarrow J$</span> is a homotopical functor of small homotopical categories, then the induced functor <span class="mathjax-tex">$f^* :\mathcal {C}^{DJ}_\mathrm {R}\rightarrow \mathcal {C}^{DI}_\mathrm {R}$</span> is exact. If <i>f</i> is injective on objects and faithful, then <span class="mathjax-tex">$f^*$</span> is a fibration.</p> <h3 class="c-article__sub-heading" id="FPar26">Proof</h3> <p>For every <span class="mathjax-tex">$x :[m] \rightarrow I$</span>, the induced functor of latching categories <span class="mathjax-tex">$\mathord \partial (DI {\downarrow }x) \rightarrow \mathord \partial (DJ {\downarrow }f x)$</span> is an isomorphism since both are essentially copies of <span class="mathjax-tex">$\mathord \partial (\Delta _\sharp {\downarrow }[m])$</span>. Moreover, if <i>f</i> is injective on objects and faithful, then <i>Df</i> is a sieve. Thus both statements follow from Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar13">1.10</a>. <span class="mathjax-tex">$\square $</span> </p> <p>For a cofibration category <span class="mathjax-tex">$\mathcal {C}$</span> we define the <i>quasicategory of frames in</i> <span class="mathjax-tex">$\mathcal {C}$</span> as a simplicial set denoted by <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> where <span class="mathjax-tex">$(\mathrm{N}_\mathrm{f}\mathcal {C})_m$</span> is the set of all homotopical Reedy cofibrant diagrams <span class="mathjax-tex">$D[m] \rightarrow \mathcal {C}$</span> ([<i>m</i>] is a homotopical category with only identities as weak equivalences). The simplicial structure is given by functoriality of <i>D</i> (using Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar25">2.1</a> to see that simplicial operators preserve Reedy cofibrancy). Since exact functors of cofibration categories preserve Reedy cofibrant diagrams, <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}$</span> is a functor from the category of cofibration categories to the category of simplicial sets.</p> <h3 class="c-article__sub-heading" id="FPar27">Remark 2.2</h3> <p>As a side note, we point out that this construction can be enhanced as follows. If <span class="mathjax-tex">$\widehat{[n]}$</span> denotes the homotopical poset [<i>n</i>] with all morphisms as weak equivalences, then the bisimplicial set</p><div id="Equ6" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}{}[m], [n] \mapsto \{ \text {homotopical Reedy cofibrant diagrams } D([m] \times \widehat{[n]}) \rightarrow \mathcal {C} \} \end{aligned}$$</span></div></div><p>is a complete Segal space with <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> as its 0th row. This enhancement will be closely analyzed in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title="Kapulkin, K., Szumiło, K.: Quasicategories of frames of cofibration categories. Appl. Categ. Struct., 1–25 (2016)" href="/article/10.1007/s40062-016-0139-x#ref-CR13" id="ref-link-section-d81634044e12438">13</a>].</p> <p>This definition can be motivated as follows. First, the objects of <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> are called <i>frames</i> in <span class="mathjax-tex">$\mathcal {C}$</span>. They are counterparts to frames in a model category <span class="mathjax-tex">$\mathcal {M}$</span>, i.e. homotopically constant Reedy cofibrant diagrams <span class="mathjax-tex">$\Delta \rightarrow \mathcal {M}$</span> which can be used to enrich the homotopy category <span class="mathjax-tex">${\text {Ho}}\mathcal {M}$</span> in the homotopy category of simplicial sets as explained in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Hovey, M.: Model categories, Mathematical Surveys and Monographs, vol. 63. American Mathematical Society, Providence, RI (1999)" href="/article/10.1007/s40062-016-0139-x#ref-CR8" id="ref-link-section-d81634044e12575">8</a>, Chapter 5]. In cofibration categories we are forced to replace <span class="mathjax-tex">$\Delta $</span> by <span class="mathjax-tex">$\Delta _\sharp $</span> since working with <span class="mathjax-tex">$\Delta $</span> would require referring to the matching objects of cosimplicial objects which are defined as certain limits and hence are not available in a cofibration category. The homotopically constant diagrams over <span class="mathjax-tex">$\Delta _\sharp $</span> are precisely the homotopical diagrams over <i>D</i>[0]. Again, one can prove using such frames that the homotopy category <span class="mathjax-tex">${\text {Ho}}\mathcal {C}$</span> is enriched in the category of homotopy types, see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Schwede, S.: The p-order of topological triangulated categories. J. Topol. 6(4), 868–914 (2013)" href="/article/10.1007/s40062-016-0139-x#ref-CR18" id="ref-link-section-d81634044e12697">18</a>, Theorems 3.10 and 3.17].<sup><a href="#Fn1"><span class="u-visually-hidden">Footnote </span>1</a></sup> Our construction can be seen as an alternative way of using frames to enrich <span class="mathjax-tex">${\text {Ho}}\mathcal {C}$</span> in homotopy types, namely, by using the mapping spaces of the quasicategory <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span>.</p><p>The second motivation is that <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> can be seen as an enhancement of the calculus of fractions. Let <span class="mathjax-tex">${\text {Sd}}[m]$</span> denote the poset of non-empty subsets of <i>m</i>. It can be seen as the full subcategory of <i>D</i>[<i>m</i>] spanned by the non-degenerate simplices of [<i>m</i>] as explained in more detail on p. 21. Homotopical Reedy cofibrant diagrams over <i>D</i>[<i>m</i>] can be seen as resolutions of their restrictions to <span class="mathjax-tex">${\text {Sd}}[m]$</span>. Therefore an object of <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> is a resolution of an object of <span class="mathjax-tex">$\mathcal {C}$</span> and a morphism is a resolution of a diagram of the form </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-v"><figure><div class="c-article-section__figure-content" id="Figv"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figv_HTML.gif?as=webp"><img aria-describedby="Figv" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figv_HTML.gif" alt="figure v" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-v-desc"></div></div></figure></div> <p>i.e. a left fraction from <span class="mathjax-tex">$X_0$</span> to <span class="mathjax-tex">$X_1$</span>. Similarly, a 2-simplex of <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> is a resolution of a diagram of the form </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-w"><figure><div class="c-article-section__figure-content" id="Figw"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figw_HTML.gif?as=webp"><img aria-describedby="Figw" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figw_HTML.gif" alt="figure w" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-w-desc"></div></div></figure></div> <p>which consists of two fractions going from <span class="mathjax-tex">$X_0$</span> to <span class="mathjax-tex">$X_1$</span> and from <span class="mathjax-tex">$X_1$</span> to <span class="mathjax-tex">$X_2$</span> along with a composite fraction going directly from <span class="mathjax-tex">$X_0$</span> to <span class="mathjax-tex">$X_2$</span>. Such diagrams simultaneously encode the composition of left fractions and the notion of equivalence of fractions. Higher simplices encode the higher homotopy of the mapping spaces of <span class="mathjax-tex">$\mathcal {C}$</span> in a similar manner.</p><p>It might be tempting to simplify the definition of <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> by replacing <i>D</i>[<i>m</i>] with <span class="mathjax-tex">${\text {Sd}}[m]$</span>. This would not work since functors <span class="mathjax-tex">${\text {Sd}}[m] \rightarrow {\text {Sd}}[n]$</span> induced by degeneracy operators <img src="//media.springernature.com/full/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_IEq405_HTML.gif" alt=""> do not respect Reedy cofibrant diagrams and thus this modification would not even yield a simplicial set.</p><p>Here is our main result. It is parametrized by a regular cardinal <span class="mathjax-tex">$\kappa $</span>. In the first two sections we will assume that <span class="mathjax-tex">$\kappa = \aleph _0$</span> to simplify the exposition.</p> <h3 class="c-article__sub-heading" id="FPar28">Theorem 2.3</h3> <p>If <span class="mathjax-tex">$\mathcal {C}$</span> is a <span class="mathjax-tex">$\kappa $</span>-cocomplete cofibration category, then <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> is a <span class="mathjax-tex">$\kappa $</span>-cocomplete quasicategory.</p> <p>Before proceeding with the proof, we will give another version of the <i>D</i> construction. For a simplicial set <i>K</i> we define a homotopical direct category <i>DK</i> as follows. The underlying category of <i>DK</i> is the category of elements of <i>K</i> but <i>only with face operators</i> as morphisms, i.e. objects of <i>DK</i> are all simplices of <i>K</i> and a morphism from <span class="mathjax-tex">$x \in K_m$</span> to <span class="mathjax-tex">$y \in K_n$</span> is an injective order preserving map <span class="mathjax-tex">$\varphi :[m] \hookrightarrow [n]$</span> such that <span class="mathjax-tex">$x = y \varphi $</span>.</p><p>Such a morphism is a <i>generating</i> weak equivalence if <span class="mathjax-tex">$y \nu $</span> is a degenerate edge of <i>K</i> where <span class="mathjax-tex">$\nu :[1] \rightarrow [n]$</span> is defined by <span class="mathjax-tex">$\nu (0) = \varphi (m)$</span> and <span class="mathjax-tex">$\nu (1) = n$</span>. The generating weak equivalences do not necessarily satisfy the 2-out-of-6 property (they are not even closed under composition in general). Thus we define the subcategory of weak equivalences as the smallest subcategory containing the generating weak equivalences and satisfying the 2-out-of-6 property. Of course, in order to verify that a functor from <i>DK</i> to a homotopical category is homotopical it suffices to check that it sends the generating weak equivalences to weak equivalences.</p><p>This construction is functorial in <i>K</i>. Moreover, the next lemma says that if <i>K</i> is the nerve of a category <i>J</i>, then <i>DK</i> coincides with <i>DJ</i> in the sense of the previous definition.</p> <h3 class="c-article__sub-heading" id="FPar29">Lemma 2.4</h3> <p>Let <i>J</i> be a category with the trivial homotopical structure. Then the homotopical categories <i>DJ</i> and <i>DNJ</i> coincide.</p> <h3 class="c-article__sub-heading" id="FPar30">Proof</h3> <p>The underlying categories of <i>DJ</i> and <i>DNJ</i> are the same by definition. The generating weak equivalences of <i>DNJ</i> are mapped to identities by <span class="mathjax-tex">$p_J :DJ \rightarrow J$</span> and hence it suffices to see that every weak equivalence created by <span class="mathjax-tex">$p_J$</span> can be obtained from the generating ones by applying the 2-out-of-6 property. Let <span class="mathjax-tex">$\varphi , \psi \in DJ$</span> and consider a morphism <span class="mathjax-tex">$\varphi \rightarrow \psi $</span> mapped by <span class="mathjax-tex">$p_J$</span> to an isomorphism <span class="mathjax-tex">$f :x \rightarrow y$</span> of <i>J</i>. Then we have a diagram </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-x"><figure><div class="c-article-section__figure-content" id="Figx"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figx_HTML.gif?as=webp"><img aria-describedby="Figx" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figx_HTML.gif" alt="figure x" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-x-desc"></div></div></figure></div> <p>in <i>DJ</i> where <i>xyxy</i> denotes the sequence </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-y"><figure><div class="c-article-section__figure-content" id="Figy"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figy_HTML.gif?as=webp"><img aria-describedby="Figy" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figy_HTML.gif" alt="figure y" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-y-desc"></div></div></figure></div> <p>and the remaining objects in the first row are its initial segments. The indicated morphisms are generating weak equivalences and hence by 2-out-of-6 <span class="mathjax-tex">$\varphi \rightarrow \psi $</span> is also a weak equivalence of <i>DNJ</i>. <span class="mathjax-tex">$\square $</span> </p> <h3 class="c-article__sub-heading" id="FPar31">Lemma 2.5</h3> <p>The functor <span class="mathjax-tex">$D :\mathsf {sSet}\rightarrow \mathsf {Cat}$</span> (i.e. when we disregard the homotopical structures of <i>DK</i>s) preserves colimits.</p> <h3 class="c-article__sub-heading" id="FPar32">Proof</h3> <p>Since <span class="mathjax-tex">$N :\mathsf {Cat}\rightarrow \mathsf {sSet}$</span> is fully faithful it reflects colimits (see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Borceux, F.: Handbook of categorical algebra. 1. Encyclopedia of Mathematics and its Applications, vol. 50, Cambridge University Press, Cambridge. Basic category theory (1994)" href="/article/10.1007/s40062-016-0139-x#ref-CR2" id="ref-link-section-d81634044e14231">2</a>, Proposition 2.2.9]). Thus it will suffice to verify that the composite functor <span class="mathjax-tex">$K \mapsto NDK$</span> preserves colimits. This follows from the fact that</p><div id="Equ7" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} (NDK)_m = \coprod _{[j_0] \hookrightarrow [j_1] \hookrightarrow \cdots \hookrightarrow [j_m]} K_{j_m}&. \end{aligned}$$</span></div></div><p> <span class="mathjax-tex">$\square $</span> </p> <p>Let <span class="mathjax-tex">$X :DK \rightarrow \mathcal {C}$</span> be a homotopical Reedy cofibrant diagram. For each simplex <span class="mathjax-tex">$x :\Delta [m] \rightarrow K$</span> consider the restriction <span class="mathjax-tex">$x^* X :D[m] \rightarrow \mathcal {C}$</span> which is an <i>m</i>-simplex of <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> (recall that <span class="mathjax-tex">$x^*$</span> is an abbreviation of <span class="mathjax-tex">$(Dx)^*$</span>). These simplices fit together to form a simplicial map <span class="mathjax-tex">$K \rightarrow \mathrm{N}_\mathrm{f}\mathcal {C}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar33">Proposition 2.6</h3> <p>Let <span class="mathjax-tex">$\mathcal {C}$</span> be a cofibration category and <i>K</i> a simplicial set. The map described above is a natural bijection between</p><ul class="u-list-style-bullet"> <li> <p>the set of homotopical Reedy cofibrant diagrams <span class="mathjax-tex">$DK \rightarrow \mathcal {C}$</span> </p> </li> <li> <p>and the set of simplicial maps <span class="mathjax-tex">$K \rightarrow \mathrm{N}_\mathrm{f}\mathcal {C}$</span>.</p> </li> </ul> <h3 class="c-article__sub-heading" id="FPar34">Proof</h3> <p>Denote the former set by <span class="mathjax-tex">$R(DK, \mathcal {C})$</span> and observe that <span class="mathjax-tex">$R(D{-}, \mathcal {C})$</span> is a contravariant functor from simplicial sets to sets. The statement says that this functor is representable and the representing object is <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span>. This will follow if we can verify that if we consider any simplicial set <i>K</i> as a colimit of its simplices, then this colimit is preserved (i.e. carried to a limit) by <span class="mathjax-tex">$R(D{-}, \mathcal {C})$</span>.</p> <p>First, note that by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar31">2.5</a> the functor <span class="mathjax-tex">$\mathsf {Cat}(D{-}, \mathcal {C})$</span> carries colimits to limits. Since <span class="mathjax-tex">$R(D{-},\mathcal {C})$</span> is a subfunctor of <span class="mathjax-tex">$\mathsf {Cat}(D{-}, \mathcal {C})$</span> it will suffice to see that a diagram <span class="mathjax-tex">$X :DK \rightarrow \mathcal {C}$</span> is homotopical and Reedy cofibrant if and only if for all <span class="mathjax-tex">$x \in K_m$</span> the induced diagram <span class="mathjax-tex">$x^* X$</span> is homotopical and Reedy cofibrant. The cofibrancy statement follows by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar13">1.10</a>.</p> <p>It is clear that if <i>X</i> is homotopical then so are all <span class="mathjax-tex">$x^* X$</span>. In order to prove the converse it suffices to consider the generating weak equivalences of <i>DK</i>. Let <span class="mathjax-tex">$x \in K_m$</span>, <span class="mathjax-tex">$y \in K_n$</span> and <span class="mathjax-tex">$\varphi :[m] \hookrightarrow [n]$</span> be such that <span class="mathjax-tex">$x = y \varphi $</span> and <span class="mathjax-tex">$y \nu $</span> is a degenerate edge where <span class="mathjax-tex">$\nu :[1] \rightarrow [n]$</span> is defined by <span class="mathjax-tex">$\nu (0) = \varphi (m)$</span> and <span class="mathjax-tex">$\nu (1) = n$</span>. We need to prove that <span class="mathjax-tex">$X \varphi $</span> is a weak equivalence in <span class="mathjax-tex">$\mathcal {C}$</span>. First, let’s assume that <span class="mathjax-tex">$\varphi (m) = n$</span>, then <span class="mathjax-tex">$\varphi $</span> is a weak equivalence when seen as a morphism <span class="mathjax-tex">$\varphi \rightarrow {\mathrm {id}}_{[n]}$</span> in <i>D</i>[<i>n</i>]. Therefore <span class="mathjax-tex">$X \varphi = (y^* X) \varphi $</span> is a weak equivalence since <span class="mathjax-tex">$y^* X$</span> is a homotopical diagram. Next, assume that <span class="mathjax-tex">$\varphi (m) < n$</span>, then <span class="mathjax-tex">$\nu $</span> is injective and can be seen as a morphism <span class="mathjax-tex">$y \nu \rightarrow y$</span> in <i>DK</i> and we have a commutative diagram on the left in <span class="mathjax-tex">$\Delta _\sharp $</span> which can be reinterpreted as a diagram in the middle in <i>DK</i> which in turn yields the diagram on the right in <span class="mathjax-tex">$\mathcal {C}$</span> (here <span class="mathjax-tex">$\varepsilon _i :[0] \rightarrow [k]$</span> is the morphism with image <i>i</i>). </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-z"><figure><div class="c-article-section__figure-content" id="Figz"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figz_HTML.gif?as=webp"><img aria-describedby="Figz" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figz_HTML.gif" alt="figure z" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-z-desc"></div></div></figure></div><p> Now, <span class="mathjax-tex">$\varepsilon _m$</span> and <span class="mathjax-tex">$\nu $</span> are weak equivalences when seen as morphisms of <i>D</i>[<i>m</i>] and <i>D</i>[<i>n</i>] respectively. Thus <span class="mathjax-tex">$X \varepsilon _m$</span> and <span class="mathjax-tex">$X \nu $</span> are weak equivalences. The edge <span class="mathjax-tex">$y \nu $</span> is degenerate, i.e. <span class="mathjax-tex">$y \nu = y \varepsilon _n \sigma _0$</span>, so the diagram <span class="mathjax-tex">$(y \nu )^* X :D[1] \rightarrow \mathcal {C}$</span> factors through <span class="mathjax-tex">$(y \varepsilon _n)^* X :D[0] \rightarrow \mathcal {C}$</span>. Since all morphisms of <i>D</i>[0] are weak equivalences it follows that <span class="mathjax-tex">$(y \nu )^* X$</span> sends all morphisms, including <span class="mathjax-tex">$\varepsilon _0$</span> above, to weak equivalences thus <span class="mathjax-tex">$X \varepsilon _0$</span> is a weak equivalence and hence so is <span class="mathjax-tex">$X \varphi $</span>. <span class="mathjax-tex">$\square $</span> </p> </div></div></section><section data-title="Reedy lifting properties"><div class="c-article-section" id="Sec4-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec4"><span class="c-article-section__title-number">4 </span>Reedy lifting properties</h2><div class="c-article-section__content" id="Sec4-content"><p>In Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s40062-016-0139-x#Sec2">1</a> we introduced <i>Reedy lifting properties</i>. Let <span class="mathjax-tex">$f :I \rightarrow J$</span> be a homotopical functor of homotopical direct categories and <span class="mathjax-tex">$F :\mathcal {C} \rightarrow \mathcal {D}$</span> an exact functor of cofibration categories. We say that <i>f</i> has the Reedy left lifting property with respect to <i>F</i> (or <i>F</i> has the Reedy right lifting property with respect to <i>f</i>) if every lifting problem </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-aa"><figure><div class="c-article-section__figure-content" id="Figaa"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figaa_HTML.gif?as=webp"><img aria-describedby="Figaa" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figaa_HTML.gif" alt="figure aa" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-aa-desc"></div></div></figure></div><p> where <i>X</i> and <i>Y</i> are homotopical Reedy cofibrant diagrams has a solution that is also a homotopical Reedy cofibrant diagram.</p><p>The previous proposition immediately implies the following.</p> <h3 class="c-article__sub-heading" id="FPar35">Corollary 3.1</h3> <p>Let <span class="mathjax-tex">$i :K \rightarrow L$</span> be a simplicial map and <span class="mathjax-tex">$F :\mathcal {C} \rightarrow \mathcal {D}$</span> an exact functor between cofibration categories. Then <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}F$</span> has the right lifting property with respect to <i>i</i> if and only if <i>F</i> has the Reedy right lifting property with respect to <i>Di</i>. <span class="mathjax-tex">$\square $</span> </p> <p>Our present goal is to verify that <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> is a <i>quasicategory</i>, i.e. that it has the right lifting property with respect to the <i>inner horn inclusions</i>, that is inclusions <span class="mathjax-tex">$\Lambda ^{i}[m] \hookrightarrow \Delta [m]$</span> for <span class="mathjax-tex">$0< i < m$</span>. To this end we employ the results of Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s40062-016-0139-x#Sec2">1</a> and verify the corresponding Reedy lifting properties of the sieves <span class="mathjax-tex">$D\Lambda ^{i}[m] \hookrightarrow D[m]$</span> induced by the inner horn inclusions. We will proceed by comparing both <i>D</i>[<i>m</i>] and <span class="mathjax-tex">$D\Lambda ^{i}[m]$</span> to [<i>m</i>] and various “generalized inner horns”.</p> <h3 class="c-article__sub-heading" id="FPar36">Lemma 3.2</h3> <p>For every <span class="mathjax-tex">$m \ge 0$</span> the functor <span class="mathjax-tex">$p_{[m]} :D[m] \rightarrow [m]$</span> is a homotopy equivalence of homotopical categories.</p> <h3 class="c-article__sub-heading" id="FPar37">Proof</h3> <p>Let <span class="mathjax-tex">$f :[m] \rightarrow D[m]$</span> be the functor that sends <span class="mathjax-tex">$i \in [m]$</span> to the standard inclusion <span class="mathjax-tex">$[i] \hookrightarrow [m]$</span>. This is a homotopical functor and we have <span class="mathjax-tex">$p_{[m]} f = {\mathrm {id}}_{[m]}$</span>. We will verify that <span class="mathjax-tex">$f p_{[m]}$</span> is weakly equivalent to <span class="mathjax-tex">${\mathrm {id}}_{D[m]}$</span> which will finish the proof.</p> <p>To this end define <span class="mathjax-tex">$s :D[m] \rightarrow D[m]$</span> as follows. Represent an object <span class="mathjax-tex">$x \in D[m]$</span> as a non-empty finite non-decreasing sequence of elements of [<i>m</i>]. Then <i>s</i>(<i>x</i>) is obtained by inserting one extra occurrence of each of the elements <span class="mathjax-tex">$0, 1, \ldots , p_{[m]}(x)$</span> into <i>x</i>. Every such element <i>i</i> is added “at the end” of the (possibly empty) block of <i>i</i>s already present in <i>x</i>. This explains the functoriality of <i>s</i>. Namely, given <span class="mathjax-tex">$\varphi :x \rightarrow y$</span> and <span class="mathjax-tex">$i \le p_{[m]}(x)$</span>, the map <span class="mathjax-tex">$s(\varphi )$</span> acts on the “old” occurrences of <i>i</i> as <span class="mathjax-tex">$\varphi $</span> does and sends the “new” occurrences to the “new” occurrences. Thus the functor <i>s</i> is homotopical and admits natural weak equivalences </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-ab"><figure><div class="c-article-section__figure-content" id="Figab"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figab_HTML.gif?as=webp"><img aria-describedby="Figab" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figab_HTML.gif" alt="figure ab" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-ab-desc"></div></div></figure></div> <p>where the map on the left inserts <i>x</i> onto the “old” occurrences in <i>s</i>(<i>x</i>) and the right one inserts <span class="mathjax-tex">$f p_{[m]}(x)$</span> onto the “new” ones. <span class="mathjax-tex">$\square $</span> </p> <p>Let <span class="mathjax-tex">$A \subseteq [m]$</span>, we define the <i>generalized horn</i> <span class="mathjax-tex">$\Lambda ^{A}[m]$</span> as the simplicial subset of <span class="mathjax-tex">$\Delta [m]$</span> generated by its codimension 1 faces containing all vertices of <i>A</i> (equivalently, codimension 1 faces lying opposite of vertices not in <i>A</i>). Observe that <span class="mathjax-tex">$\Lambda ^{\{i\}}[m] = \Lambda ^{i}[m]$</span>.</p> <h3 class="c-article__sub-heading" id="FPar38">Lemma 3.3</h3> <p>The inclusion functor <span class="mathjax-tex">$D\Lambda ^{\{1,\ldots ,m-1\}}[m] \hookrightarrow D[m]$</span> induces a weak equivalence <span class="mathjax-tex">$\mathcal {C}^{D[m]}_\mathrm {R}\rightarrow \mathcal {C}^{D\Lambda ^{\{1,\ldots ,m-1\}}[m]}_\mathrm {R}$</span> for every cofibration category <span class="mathjax-tex">$\mathcal {C}$</span> and each <span class="mathjax-tex">$m \ge 2$</span>.</p> <h3 class="c-article__sub-heading" id="FPar39">Proof</h3> <p>The functor <span class="mathjax-tex">$\mathcal {C}^{D[m]}_\mathrm {R}\rightarrow \mathcal {C}^{D\Lambda ^{\{1,\ldots ,m-1\}}[m]}_\mathrm {R}$</span> is exact by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar13">1.10</a>. By Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar7">1.7</a> (3) it suffices to verify the statement for the levelwise structures and hence it will be enough to show that the composite <span class="mathjax-tex">$D\Lambda ^{\{1,\ldots ,m-1\}}[m] \hookrightarrow D[m] \rightarrow [m]$</span> induces a weak equivalence with respect to the levelwise structures.</p> <p>In the diagram </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-ac"><figure><div class="c-article-section__figure-content" id="Figac"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figac_HTML.gif?as=webp"><img aria-describedby="Figac" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figac_HTML.gif" alt="figure ac" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-ac-desc"></div></div></figure></div> <p>the back square is a pushout of two sieves hence it induces a homotopy pullback of the associated categories of Reedy cofibrant diagrams by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar17">1.12</a>. The front square is a pushout along a sieve, but the vertical map is not a sieve. Nonetheless, the conclusion of Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar17">1.12</a> holds because of a particularly simple form of the latching categories in totally ordered sets so that a map of diagrams <span class="mathjax-tex">$[m-1] \rightarrow \mathcal {C}$</span> is a Reedy cofibration if and only if it is one when restricted along both <span class="mathjax-tex">$\delta _0$</span> and <span class="mathjax-tex">$\delta _{m-1}$</span>. Hence both squares induce homotopy pullbacks on levelwise categories of diagrams and then the assumptions of the Gluing Lemma are satisfied by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar36">3.2</a> which finishes the proof. <span class="mathjax-tex">$\square $</span> </p> <p>An <i>interval</i> is a subset of [<i>m</i>] of the form <span class="mathjax-tex">$\left\{ x \in [m]\mid i \le x \le j\right\} $</span> for some <span class="mathjax-tex">$i \le j \in [m]$</span>. In the next lemma we will consider generalized horns <span class="mathjax-tex">$\Lambda ^{A}[m]$</span> with <span class="mathjax-tex">$A \subseteq [m]$</span> such that <span class="mathjax-tex">$[m]{\setminus } A$</span> is not an interval (e.g. <span class="mathjax-tex">$A = \{ 1, \ldots , m-1 \}$</span>). Such horns are called <i>generalized inner horns</i>.</p> <h3 class="c-article__sub-heading" id="FPar40">Lemma 3.4</h3> <p>Let <span class="mathjax-tex">$A \subseteq B$</span> be subsets of [<i>m</i>] whose complements are not intervals. Then the inclusion <span class="mathjax-tex">$\Lambda ^{B}[m] \hookrightarrow \Lambda ^{A}[m]$</span> is a composite of pushouts of inner horn inclusions in dimensions at most <span class="mathjax-tex">$m - |A|$</span>. Moreover, all these horns are attached along injective maps.</p> <h3 class="c-article__sub-heading" id="FPar41">Proof</h3> <p>This follows by the proof of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Joyal, A.: The Theory of Quasi-Categories and its Applications. Quadern 45, Vol. II, Centre de Recerca Matemàtica Barcelona (2008)" href="/article/10.1007/s40062-016-0139-x#ref-CR10" id="ref-link-section-d81634044e18793">10</a>, Proposition 2.12 (iv)]. <span class="mathjax-tex">$\square $</span> </p> <p>It will now follow that <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> takes values in quasicategories. In fact, we show more. Recall that an <i>inner fibration</i> is a simplicial map with the right lifting property with respect to the inner horn inclusions.</p> <h3 class="c-article__sub-heading" id="FPar42">Proposition 3.5</h3> <p>The functor <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}$</span> carries fibrations of cofibration categories to inner fibrations. In particular, if <span class="mathjax-tex">$\mathcal {C}$</span> is a cofibration category, then <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> is a quasicategory.</p> <h3 class="c-article__sub-heading" id="FPar43">Proof</h3> <p>By Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar15">1.11</a> it suffices to check that <span class="mathjax-tex">$D\Lambda ^{i}[m] \hookrightarrow D[m]$</span> induces a weak equivalence <span class="mathjax-tex">$\mathcal {C}^{D[m]}_\mathrm {R}\rightarrow \mathcal {C}^{D\Lambda ^{i}[m]}_\mathrm {R}$</span> for every cofibration category <span class="mathjax-tex">$\mathcal {C}$</span> and <span class="mathjax-tex">$0< i < m$</span>. By Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar38">3.3</a> it will be enough to check that <span class="mathjax-tex">$D\Lambda ^{\{1,\ldots ,m-1\}}[m] \hookrightarrow D\Lambda ^{i}[m]$</span> induces a weak equivalence <span class="mathjax-tex">$\mathcal {C}^{D\Lambda ^{i}[m]}_\mathrm {R}\rightarrow \mathcal {C}^{D\Lambda ^{\{1,\ldots ,m\}}[m]}_\mathrm {R}$</span> for every cofibration category <span class="mathjax-tex">$\mathcal {C}$</span> and <span class="mathjax-tex">$0< i < m$</span>.</p> <p>That follows by an induction with respect to <i>m</i> since this inclusion is built out of pushouts of horn inclusions in dimensions below <i>m</i> by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar40">3.4</a>. Since these are pushouts along injective maps Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar17">1.12</a> says that they induce pullbacks of cofibration categories of Reedy diagrams. <span class="mathjax-tex">$\square $</span> </p> <p>In the remainder of this section we will verify some auxiliary lifting properties. First, we consider Reedy lifting properties of <span class="mathjax-tex">$[0] \hookrightarrow D{E[1]}$</span> which will be dealt with by constructing an explicit contraction of <span class="mathjax-tex">$D{E[1]}= DE(1)$</span> (here, <span class="mathjax-tex">$E(1)$</span> stands for the groupoid freely generated by one morphism <span class="mathjax-tex">$0 \rightarrow 1$</span> and <span class="mathjax-tex">${E[1]}$</span> for its nerve).</p> <h3 class="c-article__sub-heading" id="FPar44">Lemma 3.6</h3> <p>The functor <span class="mathjax-tex">$f :[0] \rightarrow DE(1)$</span> given by the sequence <span class="mathjax-tex">$0 \in DE(1)$</span> is a homotopy equivalence of homotopical categories.</p> <h3 class="c-article__sub-heading" id="FPar45">Proof</h3> <p>The proof is similar to that of Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar36">3.2</a>. This time objects of <span class="mathjax-tex">$DE(1)$</span> are represented as arbitrary finite non-empty binary sequences. Let <span class="mathjax-tex">$p :DE(1)\rightarrow [0]$</span> be the unique functor to [0] and let <span class="mathjax-tex">$s :DE(1)\rightarrow DE(1)$</span> append a new 0 to every sequence (as before, <span class="mathjax-tex">$s(\varphi )$</span> acts on “old” elements as <span class="mathjax-tex">$\varphi $</span> and sends the “new” 0 to the “new” 0). Every morphism of <i>E</i>(1) is an isomorphism so the homotopical structure on <span class="mathjax-tex">$DE(1)$</span> is the maximal one. Hence the functor <i>s</i> is homotopical and admits natural weak equivalences </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-ad"><figure><div class="c-article-section__figure-content" id="Figad"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figad_HTML.gif?as=webp"><img aria-describedby="Figad" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figad_HTML.gif" alt="figure ad" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-ad-desc"></div></div></figure></div> <p>where the map on the left inserts <i>x</i> onto the “old” occurrences in <i>s</i>(<i>x</i>) and the right one inserts <i>fp</i>(<i>x</i>) onto the “new” 0. <span class="mathjax-tex">$\square $</span> </p> <p>Before completing the main result of this section we record a corollary which considerably simplifies constructions of <span class="mathjax-tex">${E[1]}$</span>-homotopies. An <i>equivalence</i> in a quasicategory <span class="mathjax-tex">$\mathscr {C}$</span> is a morphism classified by a map <span class="mathjax-tex">$\Delta [1] \rightarrow \mathscr {C}$</span> that extends along <span class="mathjax-tex">$\Delta [1] \hookrightarrow {E[1]}$</span>. By [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Joyal, A.: The Theory of Quasi-Categories and its Applications. Quadern 45, Vol. II, Centre de Recerca Matemàtica Barcelona (2008)" href="/article/10.1007/s40062-016-0139-x#ref-CR10" id="ref-link-section-d81634044e20136">10</a>, Proposition 4.22] a morphism is a equivalence in <span class="mathjax-tex">$\mathscr {C}$</span> if and only if it becomes an isomorphism in its homotopy category.</p> <h3 class="c-article__sub-heading" id="FPar46">Corollary 3.7</h3> <p>For a cofibration category <span class="mathjax-tex">$\mathcal {C}$</span> a homotopical Reedy cofibrant diagram <span class="mathjax-tex">$X :D[1] \rightarrow \mathcal {C}$</span> is an equivalence when seen as a morphism of <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> if and only if it is homotopical with respect to <span class="mathjax-tex">$D\widehat{[1]}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar47">Proof</h3> <p>If <i>X</i> is an equivalence, then it extends to <span class="mathjax-tex">$D{E[1]}$</span>. Hence it is homotopical with respect to <span class="mathjax-tex">$D\widehat{[1]}$</span>.</p> <p>Conversely, consider a diagram </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-ae"><figure><div class="c-article-section__figure-content" id="Figae"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figae_HTML.gif?as=webp"><img aria-describedby="Figae" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figae_HTML.gif" alt="figure ae" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-ae-desc"></div></div></figure></div> <p>where the indicated maps are homotopy equivalences, the vertical ones by (the proof of) Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar36">3.2</a>, the top one by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar44">3.6</a> and the bottom one by direct inspection. Hence so is the map <span class="mathjax-tex">$D\widehat{[1]} \rightarrow D{E[1]}$</span> which is also a sieve so that the induced restriction functor <span class="mathjax-tex">$\mathcal {C}^{D{E[1]}}_\mathrm {R}\rightarrow \mathcal {C}^{D\widehat{[1]}}_\mathrm {R}$</span> is an acyclic fibration and thus every homotopical Reedy cofibrant diagram on <span class="mathjax-tex">$D\widehat{[1]}$</span> extends to one on <span class="mathjax-tex">$D{E[1]}$</span>. <span class="mathjax-tex">$\square $</span> </p> <p>An <i>isofibration</i> is a simplicial map between quasicategories with the right lifting property with respect to the inclusion <span class="mathjax-tex">$\Delta [0] \rightarrow {E[1]}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar48">Proposition 3.8</h3> <p>The functor <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}$</span> carries fibrations of cofibration categories to isofibrations.</p> <h3 class="c-article__sub-heading" id="FPar49">Proof</h3> <p>By Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar15">1.11</a> it suffices to check that <span class="mathjax-tex">$D[0] \hookrightarrow E(1)$</span> induces a weak equivalence <span class="mathjax-tex">$\mathcal {C}^{DE(1)}_\mathrm {R}\rightarrow \mathcal {C}^{D[0]}_\mathrm {R}$</span> for every cofibration category <span class="mathjax-tex">$\mathcal {C}$</span>. Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar44">3.6</a> asserts that this is the case for the composite </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-af"><figure><div class="c-article-section__figure-content" id="Figaf"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figaf_HTML.gif?as=webp"><img aria-describedby="Figaf" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figaf_HTML.gif" alt="figure af" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-af-desc"></div></div></figure></div> <p>while Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar36">3.2</a> says the same for the first functor. Thus the conclusion follows by 2-out-of-3. <span class="mathjax-tex">$\square $</span> </p> <h3 class="c-article__sub-heading" id="FPar50">Proposition 3.9</h3> <p>The functor <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}$</span> carries acyclic fibrations of cofibration categories to acyclic Kan fibrations.</p> <h3 class="c-article__sub-heading" id="FPar51">Proof</h3> <p>This follows from Lemmas <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar15">1.11</a> and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar21">1.14</a> and the fact that <span class="mathjax-tex">$D\mathord \partial \Delta [m] \hookrightarrow D[m]$</span> is a sieve for all <i>m</i>. <span class="mathjax-tex">$\square $</span> </p> <p>Before we are able to formulate the final results of this section we need to introduce <i>marked simplicial complexes</i>.</p> <h3 class="c-article__sub-heading" id="FPar52">Definition 3.10</h3> <p>A <i>marked simplicial complex</i> is a simplicial set <i>K</i> equipped with an embedding <span class="mathjax-tex">$K \hookrightarrow NP$</span> where <i>P</i> is a homotopical poset.</p> <p>Marked simplicial complexes can be seen as certain special <i>marked simplicial sets</i> which are sometimes used to provide some extra flexibility to the theory of quasicategories.</p><p>We extend the definition of <i>DK</i> to a marked simplicial complex <i>K</i> as follows. The underlying category of <i>DK</i> is the same as previously, but the homotopical structure is created by the inclusion <span class="mathjax-tex">$DK \hookrightarrow DP$</span>. This agrees with the old definition when <i>P</i> has the trivial homotopical structure.</p><p>Moreover, for a marked simplicial complex <i>K</i> we define a homotopical poset <span class="mathjax-tex">${\text {Sd}}K$</span> as the full subcategory of <i>DK</i> spanned by the non-degenerate simplices of <i>K</i> and with the homotopical structure inherited from <i>DP</i>. The category <span class="mathjax-tex">${\text {Sd}}K$</span> is known as the <i>barycentric subdivision</i> of <i>K</i> hence the notation (by analogy we may think of <i>DK</i> as the <i>fat barycentric subdivision</i> of <i>K</i>). It is indeed a poset since its objects can be identified with finite non-empty totally ordered subsets of <i>P</i> that correspond to non-degenerate simplices of <i>K</i> (just as in the classical definition of an ordered simplicial complex above) and morphisms with inclusions of such subsets. With this interpretation an inclusion <span class="mathjax-tex">$A \subseteq B$</span> is a weak equivalence if and only if <span class="mathjax-tex">$\max A \rightarrow \max B$</span> is a weak equivalence of <i>P</i> (of course, if <i>P</i> has the trivial homotopical structure, then this condition reduces to <span class="mathjax-tex">$\max A = \max B$</span>). In the case when <span class="mathjax-tex">$K = NP$</span> we will usually write <span class="mathjax-tex">${\text {Sd}}P$</span> in place of <span class="mathjax-tex">${\text {Sd}}K$</span>.</p><p>The next two lemmas will allow us to reduce constructions of diagrams over <i>DK</i> to constructions of diagrams over <span class="mathjax-tex">${\text {Sd}}K$</span>.</p> <h3 class="c-article__sub-heading" id="FPar53">Lemma 3.11</h3> <p>For any marked simplicial complex <i>K</i> the inclusion <span class="mathjax-tex">$f :{\text {Sd}}K \rightarrow DK$</span> is a homotopy equivalence.</p> <h3 class="c-article__sub-heading" id="FPar54">Proof</h3> <p>The construction is a minor modification of the one used in Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar36">3.2</a>. Let <i>P</i> denote the underlying homotopical poset of <i>K</i>. We define <span class="mathjax-tex">$q_K :DK \rightarrow {\text {Sd}}K$</span> by sending each simplex of <i>K</i> seen as a map <span class="mathjax-tex">$[k] \rightarrow P$</span> to its image and <span class="mathjax-tex">$s :DK \rightarrow DK$</span> by inserting one extra occurrence of each <span class="mathjax-tex">$p \in P$</span> that is already present in a given <span class="mathjax-tex">$x \in DK$</span>. Just as in Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar36">3.2</a> a new occurrence is inserted at the end of the block of the old occurrences which yields analogous weak equivalences </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-ag"><figure><div class="c-article-section__figure-content" id="Figag"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figag_HTML.gif?as=webp"><img aria-describedby="Figag" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figag_HTML.gif" alt="figure ag" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-ag-desc"></div></div></figure></div><p> Moreover, <span class="mathjax-tex">$q_K f = {\mathrm {id}}_{{\text {Sd}}K}$</span> which finishes the proof. <span class="mathjax-tex">$\square $</span> </p> <h3 class="c-article__sub-heading" id="FPar55">Lemma 3.12</h3> <p>Let <span class="mathjax-tex">$K \hookrightarrow L$</span> be an injective map of finite marked simplicial complexes (which means that it covers an injective homotopical map of the underlying homotopical posets). Then for every cofibration category <span class="mathjax-tex">$\mathcal {C}$</span> the inclusion <span class="mathjax-tex">$DK \cup {\text {Sd}}L \hookrightarrow DL$</span> induces an acyclic fibration <span class="mathjax-tex">$\mathcal {C}^{DL}_\mathrm {R}\rightarrow \mathcal {C}^{DK \cup {\text {Sd}}L}_\mathrm {R}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar56">Proof</h3> <p>We have the following pushout square of sieves between homotopical direct categories on the left and hence a pullback square of cofibration categories on the right by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar17">1.12</a>. </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-ah"><figure><div class="c-article-section__figure-content" id="Figah"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figah_HTML.gif?as=webp"><img aria-describedby="Figah" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figah_HTML.gif" alt="figure ah" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-ah-desc"></div></div></figure></div><p> The fibration <span class="mathjax-tex">$\mathcal {C}^{DK}_\mathrm {R}\twoheadrightarrow \mathcal {C}^{{\text {Sd}}K}_\mathrm {R}$</span> is acyclic by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar53">3.11</a> and therefore so is <span class="mathjax-tex">$\mathcal {C}^{DK \cup {\text {Sd}}L}_\mathrm {R}\twoheadrightarrow \mathcal {C}^{{\text {Sd}}L}_\mathrm {R}$</span>. Moreover, we have a triangle of fibrations </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-ai"><figure><div class="c-article-section__figure-content" id="Figai"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figai_HTML.gif?as=webp"><img aria-describedby="Figai" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figai_HTML.gif" alt="figure ai" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-ai-desc"></div></div></figure></div> <p>where <span class="mathjax-tex">$\mathcal {C}^{DL}_\mathrm {R}\twoheadrightarrow \mathcal {C}^{{\text {Sd}}L}_\mathrm {R}$</span> is acyclic again by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar53">3.11</a> and thus so is <span class="mathjax-tex">$\mathcal {C}^{DL}_\mathrm {R}\twoheadrightarrow \mathcal {C}^{DK \cup {\text {Sd}}L}_\mathrm {R}$</span>. <span class="mathjax-tex">$\square $</span> </p> <p>For future reference we will reinterpret lifting properties for special outer horns in terms of certain homotopical structures on categories <span class="mathjax-tex">$D\Lambda ^{0}[m]$</span> and <span class="mathjax-tex">$D\Lambda ^{m}[m]$</span> (special outer horns in a quasicategory <span class="mathjax-tex">$\mathscr {C}$</span> are diagrams <span class="mathjax-tex">$\Lambda ^{0}[m] \rightarrow \mathscr {C}$</span> that carry <span class="mathjax-tex">$0 \rightarrow 1$</span> to an equivalence and diagrams <span class="mathjax-tex">$\Lambda ^{m}[m] \rightarrow \mathcal {C}$</span> that carry <span class="mathjax-tex">$m-1 \rightarrow m$</span> to an equivalence).</p><p>For each <span class="mathjax-tex">$m > 1$</span> let <span class="mathjax-tex">$\langle m]$</span> denote the homotopical poset with the underlying poset [<i>m</i>] and <span class="mathjax-tex">$0 \mathop {\rightarrow }\limits ^{\sim }1$</span> as the only non-identity weak equivalence. Similarly, let <span class="mathjax-tex">$[m\rangle $</span> denote the homotopical poset with the underlying poset [<i>m</i>] and <span class="mathjax-tex">$m-1 \mathop {\rightarrow }\limits ^{\sim }m$</span> as the only non-identity weak equivalence. Let <span class="mathjax-tex">$\Lambda ^0\langle m]$</span> and <span class="mathjax-tex">$\Lambda ^{m}[m\rangle $</span> denote the outer horns seen as marked simplicial complexes with the underlying homotopical posets <span class="mathjax-tex">$\langle m]$</span> and <span class="mathjax-tex">$[m\rangle $</span>.</p> <h3 class="c-article__sub-heading" id="FPar57">Lemma 3.13</h3> <p>For every cofibration category <span class="mathjax-tex">$\mathcal {C}$</span> the inclusion <span class="mathjax-tex">$D\Lambda ^0\langle m] \hookrightarrow D\langle m]$</span> induces a weak equivalence <span class="mathjax-tex">$\mathcal {C}^{D\langle m]}_\mathrm {R}\rightarrow \mathcal {C}^{D\Lambda ^0\langle m]}_\mathrm {R}$</span>.</p> <p>The same holds for <span class="mathjax-tex">$D\Lambda ^{m}[m\rangle \hookrightarrow D[m\rangle $</span>.</p> <h3 class="c-article__sub-heading" id="FPar58">Proof</h3> <p>By Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar23">1.15</a> it will suffice to see that the inclusion <span class="mathjax-tex">$D\Lambda ^0\langle m] \hookrightarrow D\langle m]$</span> has the Reedy left lifting property with respect to all fibrations of cofibration categories.</p> <p>By Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar33">2.6</a> every Reedy lifting problem of <span class="mathjax-tex">$D\Lambda ^0\langle m] \hookrightarrow D\langle m]$</span> against a fibration of cofibration categories <span class="mathjax-tex">$P :\mathcal {C} \rightarrow \mathcal {D}$</span> is equivalent to a problem of lifting <span class="mathjax-tex">$\Lambda ^0\langle m] \hookrightarrow \langle m]$</span> against <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}P$</span> where the latter is an inner isofibration by Propositions <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar42">3.5</a> and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar48">3.8</a> and the horn is special by Corollary <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar46">3.7</a>. Hence it has a solution by [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Joyal, A.: The Theory of Quasi-Categories and its Applications. Quadern 45, Vol. II, Centre de Recerca Matemàtica Barcelona (2008)" href="/article/10.1007/s40062-016-0139-x#ref-CR10" id="ref-link-section-d81634044e23247">10</a>, Theorem 4.13].</p> <p>The same argument works for <span class="mathjax-tex">$D\Lambda ^{m}[m\rangle \hookrightarrow D[m\rangle $</span>. <span class="mathjax-tex">$\square $</span> </p> <p>We briefly recall the <i>join</i> of simplicial sets. It will be used in the proof of the next lemma and in the following section. As a functor <span class="mathjax-tex">$\star :\Delta \times \Delta \rightarrow \Delta $</span> it is defined by concatenation: <span class="mathjax-tex">$[m], [n] \mapsto [m + 1 + n]$</span>. Then the general join is defined as the unique functor <span class="mathjax-tex">$\mathsf {sSet}\times \mathsf {sSet}\rightarrow \mathsf {sSet}$</span> which agrees with the above on the representable simplicial sets and such that for each <i>K</i> the resulting functor <span class="mathjax-tex">$K \star {-}:\mathsf {sSet}\rightarrow K {\downarrow }\mathsf {sSet}$</span> preserves colimits. More explicitly, the join of simplicial sets <i>K</i> and <i>L</i> can be described as</p><div id="Equ8" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} (K \star L)_p = \coprod _{m + 1 + n = p} K_m \times L_n \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$m, n \ge -1$</span> and <span class="mathjax-tex">$K_{-1}$</span> and <span class="mathjax-tex">$L_{-1}$</span> are understood to be one element sets.</p><p>The functor <span class="mathjax-tex">$K \star {-}:\mathsf {sSet}\rightarrow K {\downarrow }\mathsf {sSet}$</span> has a right adjoint. Its value at an object <span class="mathjax-tex">$X :K \rightarrow L$</span> is called the <i>slice of X under L</i> and denoted by <span class="mathjax-tex">$X {\backslash }L$</span>. See [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="Joyal, A.: Quasi-categories and Kan complexes. J. Pure Appl. Algebra 175(1–3), 207–222 (2002). doi:
 10.1016/S0022-4049(02)00135-4
 
 . Special volume celebrating the 70th birthday of Professor Max Kelly. MR1935979" href="/article/10.1007/s40062-016-0139-x#ref-CR9" id="ref-link-section-d81634044e23815">9</a>, Section 3] for details.</p><p>Let <span class="mathjax-tex">$[k + \widetilde{1} + m]$</span> denote a homotopical category with underlying category <span class="mathjax-tex">$[k+1+m]$</span> and <span class="mathjax-tex">$k \mathop {\rightarrow }\limits ^{\sim }k+1$</span> as the only non-identity weak equivalence. Let <span class="mathjax-tex">$\Lambda ^{[k]}[k+\widetilde{1}+m]$</span> denote the generalized horn <span class="mathjax-tex">$\Lambda ^{[k]}[k+1+m]$</span> seen as a marked simplicial complex with the underlying homotopical poset <span class="mathjax-tex">$[k+\widetilde{1}+m]$</span>. The next lemma is a generalization of the previous one.</p> <h3 class="c-article__sub-heading" id="FPar59">Lemma 3.14</h3> <p>The inclusion <span class="mathjax-tex">$D\Lambda ^{[k]}[k+\widetilde{1}+m] \hookrightarrow D[k+\widetilde{1}+m]$</span> has the Reedy left lifting property with respect to all fibrations of cofibration categories. Hence for any cofibration category <span class="mathjax-tex">$\mathcal {C}$</span> it induces a weak equivalence <span class="mathjax-tex">$\mathcal {C}^{D[k+\widetilde{1}+m]}_\mathrm {R}\rightarrow \mathcal {C}^{D\Lambda ^{[k]}[k+\widetilde{1}+m]}_\mathrm {R}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar60">Proof</h3> <p>The case of <span class="mathjax-tex">$k = 0$</span> is just the previous lemma (with <i>m</i> replaced by <span class="mathjax-tex">$1+m$</span>). The case of <span class="mathjax-tex">$k > 0$</span> can be reduced to the case of <span class="mathjax-tex">$k = 0$</span> as follows. We have <span class="mathjax-tex">$[k+1+m] \cong [k] \star [m]$</span> and <span class="mathjax-tex">$\Lambda ^{[k]}[k+1+m] \cong \Delta [k] \star \mathord \partial \Delta [m]$</span> and hence it will suffice to solve every lifting problem </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-aj"><figure><div class="c-article-section__figure-content" id="Figaj"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figaj_HTML.gif?as=webp"><img aria-describedby="Figaj" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figaj_HTML.gif" alt="figure aj" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-aj-desc"></div></div></figure></div> <p>where <i>X</i> and <i>Y</i> send the edge <span class="mathjax-tex">$k \rightarrow k+1$</span> to an equivalence and <i>P</i> is an inner isofibration (by Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar33">2.6</a>). This problem is equivalent to </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-ak"><figure><div class="c-article-section__figure-content" id="Figak"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figak_HTML.gif?as=webp"><img aria-describedby="Figak" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figak_HTML.gif" alt="figure ak" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-ak-desc"></div></div></figure></div> <p>where <span class="mathjax-tex">$X'$</span> and <span class="mathjax-tex">$Y'$</span> are the restrictions of <i>X</i> and <i>Y</i> to <span class="mathjax-tex">$\Delta [k-1]$</span> so that the resulting horn is special (under identifications <span class="mathjax-tex">$\{ k \} \star \Delta [m] \cong \Delta [1+m]$</span> and <span class="mathjax-tex">$\{ k \} \star \mathord \partial \Delta [m] \cong \Lambda ^{0}[1+m]$</span>). The map <span class="mathjax-tex">$X' {\backslash }\mathscr {C} \rightarrow Y' {\backslash }\mathscr {D}$</span> is an inner isofibration between quasicategories by [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Joyal, A.: The Theory of Quasi-Categories and its Applications. Quadern 45, Vol. II, Centre de Recerca Matemàtica Barcelona (2008)" href="/article/10.1007/s40062-016-0139-x#ref-CR10" id="ref-link-section-d81634044e24958">10</a>, Theorem 3.19]. Thus a solution exists by the case of <span class="mathjax-tex">$k = 0$</span>. <span class="mathjax-tex">$\square $</span> </p> <p>We conclude this section with a technical observation about limits of cofibration categories. Corollary <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar63">3.16</a>.1 below is a version of Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar17">1.12</a>.</p> <h3 class="c-article__sub-heading" id="FPar61">Lemma 3.15</h3> <p>Let <img src="//media.springernature.com/full/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_IEq720_HTML.gif" alt=""> be a surjective simplicial map. Then for every cofibration category <span class="mathjax-tex">$\mathcal {C}$</span>, a diagram <span class="mathjax-tex">$X :D L \rightarrow \mathcal {C}$</span> is Reedy cofibrant if and only if <span class="mathjax-tex">$f^* X$</span> is.</p> <h3 class="c-article__sub-heading" id="FPar62">Proof</h3> <p>For a simplex <span class="mathjax-tex">$y \in L_m$</span> pick a simplex <span class="mathjax-tex">$x \in K_m$</span> such that <span class="mathjax-tex">$f x = y$</span>. Then the induced functor of latching categories <span class="mathjax-tex">$\mathord \partial (D K {\downarrow }x) \rightarrow \mathord \partial (D L {\downarrow }y)$</span> is an isomorphism since both are essentially copies of <span class="mathjax-tex">$\mathord \partial (\Delta _\sharp {\downarrow }[m])$</span>. Thus the latching map of <i>K</i> at <i>x</i> is a cofibration if and only if the latching map of <i>L</i> at <i>y</i> is. <span class="mathjax-tex">$\square $</span> </p> <h3 class="c-article__sub-heading" id="FPar63">Corollary 3.16</h3> <ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(1)</span> <p>If </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-al"><figure><div class="c-article-section__figure-content" id="Figal"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figal_HTML.gif?as=webp"><img aria-describedby="Figal" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figal_HTML.gif" alt="figure al" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-al-desc"></div></div></figure></div><p> is a pushout square of simplicial sets (along a monomorphism <span class="mathjax-tex">$A \hookrightarrow B$</span>), then the square of cofibration categories </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-am"><figure><div class="c-article-section__figure-content" id="Figam"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figam_HTML.gif?as=webp"><img aria-describedby="Figam" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figam_HTML.gif" alt="figure am" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-am-desc"></div></div></figure></div><p> is a pullback.</p> </li> <li> <span class="u-custom-list-number">(2)</span> <p>If a simplicial set <i>K</i> is a colimit of a sequence of monomorphisms </p><div id="Equ9" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} K_0 \hookrightarrow K_1 \hookrightarrow K_2 \hookrightarrow \cdots \text {,} \end{aligned}$$</span></div></div><p> then <span class="mathjax-tex">$\mathcal {C}^{D K}_\mathrm {R}$</span> is the limit of the tower of fibrations </p><div id="Equ10" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \cdots \twoheadrightarrow \mathcal {C}^{D K_2}_\mathrm {R}\twoheadrightarrow \mathcal {C}^{D K_1}_\mathrm {R}\twoheadrightarrow \mathcal {C}^{D K_0}_\mathrm {R}\text {.} \end{aligned}$$</span></div></div> </li> </ol> <h3 class="c-article__sub-heading" id="FPar64">Proof</h3> <p>We verify the first statement, the proof of the second one is similar.</p> <p>By Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar31">2.5</a>, it is enough to verify that a diagram <span class="mathjax-tex">$X :D L \rightarrow \mathcal {C}$</span> is homotopical Reedy cofibrant if and only if its restrictions to <i>DB</i> and <i>DK</i> are. For cofibrancy, it follows by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar61">3.15</a> applied to the surjection <img src="//media.springernature.com/full/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_IEq733_HTML.gif" alt="">.</p> <p>For the other part, assume that <i>X</i> | <i>DB</i> and <i>X</i> | <i>DK</i> are homotopical. Take <span class="mathjax-tex">$y \in L_n$</span> and <span class="mathjax-tex">$\varphi :[m] \hookrightarrow [n]$</span> such that <span class="mathjax-tex">$y \varphi \rightarrow y$</span> is a generating weak equivalence in <i>DL</i> (i.e. <span class="mathjax-tex">$y \nu $</span> is a degenerate edge of <i>L</i> where <span class="mathjax-tex">$\nu :[1] \rightarrow [n]$</span> is defined by <span class="mathjax-tex">$\nu (0) = \varphi (m)$</span> and <span class="mathjax-tex">$\nu (1) = n$</span>). It will be enough to verify that it arises from a weak equivalence in <i>DB</i> or <i>DK</i>. By 2-out-of-6 we can assume that <span class="mathjax-tex">$m = 0$</span>, <span class="mathjax-tex">$n = 1$</span>, <span class="mathjax-tex">$\varphi =\delta _1$</span> and <span class="mathjax-tex">$\nu = {\mathrm {id}}_{[1]}$</span> so that <span class="mathjax-tex">$y \nu = y$</span> (if <span class="mathjax-tex">$\nu $</span> is not injective, the conclusion is immediate). We consider two cases.</p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(1)</span> <p>If <i>y</i> is a simplex of <i>K</i>, it is also degenerate in <i>K</i> since <span class="mathjax-tex">$K \rightarrow L$</span> is injective. Thus <span class="mathjax-tex">$y \varphi \rightarrow y$</span> is a weak equivalence in <i>DK</i>.</p> </li> <li> <span class="u-custom-list-number">(2)</span> <p>If <i>y</i> is not a simplex of <i>K</i>, then it comes from a unique simplex <i>x</i> of <i>B</i>. This simplex is also degenerate since otherwise it would have more distinct degeneracies than <i>y</i> and the square would not be a pushout. In this case, <span class="mathjax-tex">$x \varphi \rightarrow x$</span> is a weak equivalence in <i>DB</i>. <span class="mathjax-tex">$\square $</span> </p> </li> </ol> </div></div></section><section data-title="Cocompleteness: the infinite case"><div class="c-article-section" id="Sec5-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec5"><span class="c-article-section__title-number">5 </span>Cocompleteness: the infinite case</h2><div class="c-article-section__content" id="Sec5-content"><p>In the last two sections we will verify that <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}$</span> takes values in <span class="mathjax-tex">$\kappa $</span>-cocomplete quasicategories and <span class="mathjax-tex">$\kappa $</span>-cocontinuous functors. From this point on the cases of finitely cocomplete cofibration categories and <span class="mathjax-tex">$\kappa $</span>-cocomplete cofibration categories for <span class="mathjax-tex">$\kappa > \aleph _0$</span> will diverge. The general approaches to both cases are still analogous, but they differ in technical details and there seems to be no way of presenting them in a completely uniform manner. The presence of infinite homotopy colimits allows us to use simpler constructions so we will consider the case of <span class="mathjax-tex">$\kappa > \aleph _0$</span> first (so in this section, <span class="mathjax-tex">$\mathcal {C}$</span> will denote a <span class="mathjax-tex">$\kappa $</span>-cocomplete cofibration category with <span class="mathjax-tex">$\kappa > \aleph _0$</span>, see Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s40062-016-0139-x#Sec2">1</a>). The remaining case of <span class="mathjax-tex">$\kappa = \aleph _0$</span> will be covered in the next section.</p><p>First, we briefly review colimits in quasicategories. Given a simplicial set <i>K</i>, we denote <span class="mathjax-tex">$K \star \Delta [0]$</span> by <span class="mathjax-tex">$K^\rhd $</span> and call it the <i>cone</i> under <i>K</i>. Given a quasicategory <span class="mathjax-tex">$\mathscr {C}$</span> and a diagram <span class="mathjax-tex">$X :K \rightarrow \mathscr {C}$</span>, any extension of <i>X</i> to <span class="mathjax-tex">$K^\rhd $</span> is called a <i>cone</i> under <i>X</i>. Such a cone <i>S</i> is <i>universal</i> or a <i>colimit</i> of <i>X</i> if for any <span class="mathjax-tex">$m > 0$</span> and any diagram of solid arrows </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-an"><figure><div class="c-article-section__figure-content" id="Figan"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figan_HTML.gif?as=webp"><img aria-describedby="Figan" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figan_HTML.gif" alt="figure an" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-an-desc"></div></div></figure></div> <p>where <span class="mathjax-tex">$U|K^\rhd = S$</span> there exists a dashed arrow making the diagram commute. Given a regular cardinal <span class="mathjax-tex">$\kappa $</span>, we say that <span class="mathjax-tex">$\mathscr {C}$</span> is <span class="mathjax-tex">$\kappa $</span> <i>-cocomplete</i> all diagrams in <span class="mathjax-tex">$\mathscr {C}$</span> indexed over <span class="mathjax-tex">$\kappa $</span>-small simplicial sets admit colimits.</p><p>Before we show that <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> is such a quasicategory, we need a few preliminary lemmas. If <i>I</i> is a discrete category, then colimits over <span class="mathjax-tex">$[0] \star I$</span> are called wide pushouts. A wide pushout of a diagram <span class="mathjax-tex">$X :[0] \star I \rightarrow \mathcal {C}$</span> will be denoted by</p><div id="Equ41" class="c-article-equation"><div class="c-article-equation__content"><img src="//media.springernature.com/full/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Equ41_HTML.gif" class="u-display-block" alt=""></div></div><p>In order to understand colimits in <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span>, we need to analyze diagrams over direct categories of the form <span class="mathjax-tex">$D(K \star \Delta [m])$</span> </p> <h3 class="c-article__sub-heading" id="FPar65">Lemma 4.1</h3> <p>Let <span class="mathjax-tex">$\mathcal {C}$</span> be a <span class="mathjax-tex">$\kappa $</span>-cocomplete cofibration category and <i>K</i> a <span class="mathjax-tex">$\kappa $</span>-small simplicial set. If <span class="mathjax-tex">$X :D(K \star \Delta [m]) \rightarrow \mathcal {C}$</span> is a homotopical Reedy cofibrant diagram, then the induced morphism</p><div id="Equ11" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} X_{[m]} \rightarrow {\text {colim}}_{D(K \star \Delta [m])} X \end{aligned}$$</span></div></div><p>is a weak equivalence.</p> <h3 class="c-article__sub-heading" id="FPar66">Proof</h3> <p>The morphism in question factors as</p><div id="Equ12" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} X_{[m]} \rightarrow {\text {colim}}_{D[m]} X \rightarrow {\text {colim}}_{D(K \star \Delta [m])} X \end{aligned}$$</span></div></div><p>where the first morphism is a weak equivalence by [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title="Kapulkin, K., Szumiło, K.: Quasicategories of frames of cofibration categories. Appl. Categ. Struct., 1–25 (2016)" href="/article/10.1007/s40062-016-0139-x#ref-CR13" id="ref-link-section-d81634044e27494">13</a>, Lemma 3.17] and Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar36">3.2</a>. Thus it will be enough to check that the second one is.</p> <p>It will suffice to verify that this statement holds when <i>K</i> is a simplex and that it is preserved under coproducts, pushouts along monomorphisms and colimits of sequences of monomorphisms.</p> <p>Let <span class="mathjax-tex">$K = \Delta [k]$</span> and let <span class="mathjax-tex">$\iota $</span> be the composite <span class="mathjax-tex">$[m] \hookrightarrow [k] \star [m] \cong [k+1+m]$</span>. Then we have a commutative square </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-ao"><figure><div class="c-article-section__figure-content" id="Figao"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figao_HTML.gif?as=webp"><img aria-describedby="Figao" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figao_HTML.gif" alt="figure ao" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-ao-desc"></div></div></figure></div> <p>where the left morphism is a weak equivalence since <i>X</i> is homotopical and so are the horizontal ones by the argument above. Thus the right morphism is also a weak equivalence.</p> <p>Next, consider a pushout square </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-ap"><figure><div class="c-article-section__figure-content" id="Figap"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figap_HTML.gif?as=webp"><img aria-describedby="Figap" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figap_HTML.gif" alt="figure ap" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-ap-desc"></div></div></figure></div> <p>such that the statement holds for <i>A</i>, <i>B</i> and <i>K</i>. The functor <span class="mathjax-tex">${-}\star \Delta [m]$</span> preserves pushouts and so does <i>D</i> by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar31">2.5</a>. Thus in the cube </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-aq"><figure><div class="c-article-section__figure-content" id="Figaq"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figaq_HTML.gif?as=webp"><img aria-describedby="Figaq" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figaq_HTML.gif" alt="figure aq" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-aq-desc"></div></div></figure></div> <p>both the front and the back faces are pushouts along sieves and the conclusion follows by [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Răadulescu-Banu, A.: Cofibrations in Homotopy Theory (2006). 
 http://arxiv.org/abs/math/0610009v4
 
 
 " href="/article/10.1007/s40062-016-0139-x#ref-CR14" id="ref-link-section-d81634044e27727">14</a>, Theorem 9.4.1 (1a)] and the Gluing Lemma (here, we also use the fact that a colimit of a diagram whose indexing category is a colimit of a diagram of categories can be computed as an iterated colimit).</p> <p>The case of colimits of sequences of monomorphisms is similar and we omit it.</p> <p>The case of coproducts is also similar, but there is a difference in the fact that <span class="mathjax-tex">${-}\star \Delta [m]$</span> does not preserve coproducts. Instead, it sends coproducts to wide pushouts under <span class="mathjax-tex">$\Delta [m]$</span>. Thus if we have a <span class="mathjax-tex">$\kappa $</span>-small family <span class="mathjax-tex">$\{ K_i \mid i \in I \}$</span> of <span class="mathjax-tex">$\kappa $</span>-small simplicial sets and a diagram <span class="mathjax-tex">$X :D((\coprod _i K_i) \star \Delta [m]) \rightarrow \mathcal {C}$</span>, then there is a canonical isomorphism</p><div id="Equ42" class="c-article-equation"><div class="c-article-equation__content"><img src="//media.springernature.com/full/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Equ42_HTML.gif" class="u-display-block" alt=""></div></div><p>The conclusion follows by the fact that in a cofibration category all the structure morphisms of a wide pushout of acyclic cofibrations are weak equivalences (by [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title="Kapulkin, K., Szumiło, K.: Quasicategories of frames of cofibration categories. Appl. Categ. Struct., 1–25 (2016)" href="/article/10.1007/s40062-016-0139-x#ref-CR13" id="ref-link-section-d81634044e27972">13</a>, Lemma 3.17] since <span class="mathjax-tex">$\widehat{[0] \star I}$</span> is contractible to its cone object as a homotopical category). <span class="mathjax-tex">$\square $</span> </p> <p>Note that for any simplicial set <i>K</i> there is a unique functor <span class="mathjax-tex">$p_K :D(K^\rhd ) \rightarrow (DK)^\rhd $</span> that restricts to the identity of <i>DK</i> and sends all the objects not in <i>DK</i> to the cone point of <span class="mathjax-tex">$(DK)^\rhd $</span>. This functor is homotopical. In the next lemma we use it to compare colimits over <i>DK</i> and <span class="mathjax-tex">$D(K^\rhd )$</span>.</p> <h3 class="c-article__sub-heading" id="FPar67">Lemma 4.2</h3> <p>Let <span class="mathjax-tex">$\mathcal {C}$</span> be a <span class="mathjax-tex">$\kappa $</span>-cocomplete cofibration category, <i>K</i> a <span class="mathjax-tex">$\kappa $</span>-small simplicial set and <span class="mathjax-tex">$X :DK \rightarrow \mathcal {C}$</span> a homotopical Reedy cofibrant diagram. Consider a morphism <span class="mathjax-tex">$f :{\text {colim}}_{DK} X \rightarrow Y$</span> and the corresponding cone <span class="mathjax-tex">$\widetilde{T} :(DK)^\rhd \rightarrow \mathcal {C}$</span>. If <i>T</i> is any Reedy cofibrant replacement of <span class="mathjax-tex">$p_K^* \widetilde{T}$</span> relative to <i>DK</i> (which exists by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar11">1.9</a>), then <i>f</i> factors as</p><div id="Equ13" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\text {colim}}_{DK} X \rightarrow {\text {colim}}_{D(K^\rhd )} T \mathop {\rightarrow }\limits ^{\sim }Y \text {.} \end{aligned}$$</span></div></div> <h3 class="c-article__sub-heading" id="FPar68">Proof</h3> <p>To verify that the above composite agrees with <i>f</i> it suffices to check that it agrees upon precomposition with <span class="mathjax-tex">$X_x \rightarrow {\text {colim}}_{DK} X$</span> for all <span class="mathjax-tex">$x \in DK$</span>. That’s indeed the case since <span class="mathjax-tex">$T|DK = X$</span>.</p> <p>It remains to check that the latter morphism is a weak equivalence. In the diagram </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-ar"><figure><div class="c-article-section__figure-content" id="Figar"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figar_HTML.gif?as=webp"><img aria-describedby="Figar" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figar_HTML.gif" alt="figure ar" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-ar-desc"></div></div></figure></div> <p>the left morphism is a weak equivalence by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar65">4.1</a> and so is the diagonal one since <i>T</i> is a cofibrant replacement of <span class="mathjax-tex">$p_K^* \widetilde{T}$</span>. Therefore the top morphism is also a weak equivalence. <span class="mathjax-tex">$\square $</span> </p> <p>We will need an augmented version of the <i>D</i> construction. In fact, we will only need to apply it to [<i>m</i>] and <span class="mathjax-tex">$\mathord \partial \Delta [m]$</span> so we define it only in these cases.</p><p>We will denote by <span class="mathjax-tex">$D_\mathrm {a}[m]$</span> the category of all order preserving maps <span class="mathjax-tex">$[k] \rightarrow [m]$</span> including the one with <span class="mathjax-tex">$[k] = [-1] = \varnothing $</span>. A morphism from <span class="mathjax-tex">$x :[k] \rightarrow [m]$</span> to <span class="mathjax-tex">$y :[l] \rightarrow [m]$</span> is an injective order preserving map <span class="mathjax-tex">$\varphi :[k] \hookrightarrow [l]$</span> such that <span class="mathjax-tex">$x = y \varphi $</span>. In other words, <span class="mathjax-tex">$D_\mathrm {a}[m]$</span> is obtained from <i>D</i>[<i>m</i>] by adjoining an initial object. The homotopical structure on <span class="mathjax-tex">$D_\mathrm {a}[m]$</span> is an extension of the one on <i>D</i>[<i>m</i>] where <span class="mathjax-tex">$[-1] \rightarrow [m]$</span> is not weakly equivalent to any other object. We will also consider a slightly richer homotopical structure <span class="mathjax-tex">$\widetilde{D}_\mathrm {a}[m]$</span> where <span class="mathjax-tex">$[-1] \rightarrow [m]$</span> is weakly equivalent to all the constant maps with the value 0.</p><p>The homotopical categories <span class="mathjax-tex">$D_\mathrm {a}\mathord \partial \Delta [m]$</span> and <span class="mathjax-tex">$\widetilde{D}_\mathrm {a}\mathord \partial \Delta [m]$</span> are the full homotopical subcategories of <span class="mathjax-tex">$D_\mathrm {a}[m]$</span> and <span class="mathjax-tex">$\widetilde{D}_\mathrm {a}[m]$</span> spanned by the non-surjective maps <span class="mathjax-tex">$[k] \rightarrow [m]$</span> [i.e. by the simplices of <span class="mathjax-tex">$\mathord \partial \Delta [m]$</span> including the “<span class="mathjax-tex">$(-1)$</span>-dimensional” one].</p><p>Similarly, the homotopical posets <span class="mathjax-tex">${\text {Sd}}_\mathrm {a}[m]$</span>, <span class="mathjax-tex">$\widetilde{\text {Sd}}_\mathrm {a}[m]$</span>, <span class="mathjax-tex">${\text {Sd}}_\mathrm {a}\mathord \partial \Delta [m]$</span> and <span class="mathjax-tex">$\widetilde{\text {Sd}}_\mathrm {a}\mathord \partial \Delta [m]$</span> are the full homotopical subcategories of <span class="mathjax-tex">$D_\mathrm {a}[m]$</span>, <span class="mathjax-tex">$\widetilde{D}_\mathrm {a}[m]$</span>, <span class="mathjax-tex">$D_\mathrm {a}\mathord \partial \Delta [m]$</span> and <span class="mathjax-tex">$\widetilde{D}_\mathrm {a}\mathord \partial \Delta [m]$</span> respectively spanned by their objects that are injective as maps <span class="mathjax-tex">$[k] \rightarrow [m]$</span>.</p> <h3 class="c-article__sub-heading" id="FPar69">Lemma 4.3</h3> <p>The restriction functors</p><div id="Equ43" class="c-article-equation"><div class="c-article-equation__content"><img src="//media.springernature.com/full/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Equ43_HTML.gif" class="u-display-block" alt=""></div></div><p>are all acyclic fibrations.</p> <h3 class="c-article__sub-heading" id="FPar70">Proof</h3> <p>All these functors are induced by sieves so they are fibrations. We will construct a homotopy inverse to <span class="mathjax-tex">$f :\widetilde{\text {Sd}}_\mathrm {a}[m] \hookrightarrow \widetilde{D}_\mathrm {a}[m]$</span> which will restrict to homotopy inverses of all the other sieves in question. The conclusion will follow by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar9">1.8</a> (3). The construction is a minor modification of the one used in Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar36">3.2</a> (and essentially the same as in Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar53">3.11</a>). Namely, we define <span class="mathjax-tex">$q :\widetilde{D}_\mathrm {a}[m] \rightarrow \widetilde{\text {Sd}}_\mathrm {a}[m]$</span> by sending each <span class="mathjax-tex">$[k] \rightarrow [m]$</span> to its image and <span class="mathjax-tex">$s :\widetilde{D}_\mathrm {a}[m] \rightarrow \widetilde{D}_\mathrm {a}[m]$</span> by inserting one extra occurrence of each <span class="mathjax-tex">$i \in [m]$</span> that is already present in a given <span class="mathjax-tex">$x \in \widetilde{D}_\mathrm {a}[m]$</span>. Just as in Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar36">3.2</a> a new occurrence is inserted at the end of the block of the old occurrences which yields analogous weak equivalences </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-as"><figure><div class="c-article-section__figure-content" id="Figas"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figas_HTML.gif?as=webp"><img aria-describedby="Figas" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figas_HTML.gif" alt="figure as" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-as-desc"></div></div></figure></div><p> Moreover, <span class="mathjax-tex">$q f = {\mathrm {id}}_{\widetilde{\text {Sd}}_\mathrm {a}[m]}$</span> which finishes the proof. <span class="mathjax-tex">$\square $</span> </p> <p>Homotopical Reedy cofibrant diagrams on <span class="mathjax-tex">$D_\mathrm {a}[1]$</span> will be used to encode cones on diagrams in <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> and the ones which are homotopical with respect to <span class="mathjax-tex">$\widetilde{D}_\mathrm {a}[1]$</span> will correspond to the universal cones. The following lemma (and, more directly, Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar73">4.5</a> below) will translate between the universality of such cones in <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> and strict colimits of the corresponding diagrams in <span class="mathjax-tex">$\mathcal {C}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar71">Lemma 4.4</h3> <p>The two functors</p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(1)</span> <p> <span class="mathjax-tex">$\mathcal {C}^{\widetilde{\text {Sd}}_\mathrm {a}[m]}_\mathrm {R}\rightarrow \mathcal {C}^{\widetilde{\text {Sd}}_\mathrm {a}\mathord \partial \Delta [m]}_\mathrm {R}$</span> and</p> </li> <li> <span class="u-custom-list-number">(2)</span> <p> <span class="mathjax-tex">$\mathcal {C}^{\widetilde{D}_\mathrm {a}[m]}_\mathrm {R}\rightarrow \mathcal {C}^{\widetilde{D}_\mathrm {a}\mathord \partial \Delta [m]}_\mathrm {R}$</span> </p> </li> </ol><p>induced by the inclusion <span class="mathjax-tex">$\mathord \partial \Delta [m] \hookrightarrow \Delta [m]$</span> are acyclic fibrations.</p> <h3 class="c-article__sub-heading" id="FPar72">Proof</h3> <p>Both inclusions <span class="mathjax-tex">$\widetilde{\text {Sd}}_\mathrm {a}\mathord \partial \Delta [m] \hookrightarrow \widetilde{\text {Sd}}_\mathrm {a}[m]$</span> and <span class="mathjax-tex">$\widetilde{D}_\mathrm {a}\mathord \partial \Delta [m] \hookrightarrow \widetilde{D}_\mathrm {a}[m]$</span> are sieves hence by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar9">1.8</a> (3) it will be enough to prove that they are homotopy equivalences.</p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(1)</span> <p>Consider two homotopical functors <span class="mathjax-tex">$i_0, i_1 :{\text {Sd}}_\mathrm {a}[m-1] \rightarrow \widetilde{\text {Sd}}_\mathrm {a}[m]$</span> defined as <span class="mathjax-tex">$i_0 A = \left\{ k + 1\mid k \in A\right\} $</span> and <span class="mathjax-tex">$i_1 A = i_0 A \cup \{ 0 \}$</span> for any <span class="mathjax-tex">$A \subseteq [m-1]$</span>. We have <span class="mathjax-tex">$i_0 A \subseteq i_1 A$</span> and the resulting natural transformation induces an isomorphism of homotopical categories <span class="mathjax-tex">${\text {Sd}}_\mathrm {a}[m-1] \times \widehat{[1]} \rightarrow \widetilde{\text {Sd}}_\mathrm {a}[m]$</span>. It follows that <span class="mathjax-tex">$i_0$</span> is a homotopy equivalence since <span class="mathjax-tex">$[0] \hookrightarrow \widehat{[1]}$</span> is. This homotopy equivalence also restricts to a homotopy equivalence <span class="mathjax-tex">${\text {Sd}}_\mathrm {a}[m-1] \hookrightarrow \widetilde{\text {Sd}}_\mathrm {a}\mathord \partial \Delta [m]$</span> and thus the conclusion follows by the triangle </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-at"><figure><div class="c-article-section__figure-content" id="Figat"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figat_HTML.gif?as=webp"><img aria-describedby="Figat" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figat_HTML.gif" alt="figure at" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-at-desc"></div></div></figure></div> </li> <li> <span class="u-custom-list-number">(2)</span> <p>We have a square </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-au"><figure><div class="c-article-section__figure-content" id="Figau"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figau_HTML.gif?as=webp"><img aria-describedby="Figau" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figau_HTML.gif" alt="figure au" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-au-desc"></div></div></figure></div><p> where the top functor is a homotopy equivalence by the first part of the lemma and so are the vertical ones by the proof of Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar69">4.3</a>. Therefore so is the bottom one. <span class="mathjax-tex">$\square $</span> </p> </li> </ol> <p>For every <span class="mathjax-tex">$m > 0$</span> each object of <span class="mathjax-tex">$D(K \star \Delta [m])$</span> can be uniquely written as <span class="mathjax-tex">$x \star \varphi $</span> with <span class="mathjax-tex">$x \in D_\mathrm {a}K$</span> and <span class="mathjax-tex">$\varphi \in D_\mathrm {a}[m]$</span>. This yields a functor <span class="mathjax-tex">$r_K :D(K \star \Delta [m]) \rightarrow D_\mathrm {a}[m]$</span> sending <span class="mathjax-tex">$x \star \varphi $</span> to <span class="mathjax-tex">$\varphi $</span> to which we associate the left Kan extension</p><div id="Equ14" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\text {Lan}}_{r_K} :\mathcal {C}^{D(K \star \Delta [m])}_\mathrm {R}\rightarrow \mathcal {C}^{D_\mathrm {a}[m]}_\mathrm {R}\end{aligned}$$</span></div></div><p>which can be constructed as</p><div id="Equ15" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} ({\text {Lan}}_{r_K} X)_\varphi = {\text {colim}}_{D[k]} \varphi ^* X \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$\varphi :[k] \rightarrow [m]$</span> (this colimit exists since <span class="mathjax-tex">$\varphi ^* X$</span> is Reedy cofibrant by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar25">2.1</a>). Analogously, we have a functor <span class="mathjax-tex">$s_K :D(K \star \mathord \partial \Delta [m]) \rightarrow D_\mathrm {a}\mathord \partial \Delta [m]$</span> and the associated left Kan extension</p><div id="Equ16" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\text {Lan}}_{s_K} :\mathcal {C}^{D(K \star \mathord \partial \Delta [m])}_\mathrm {R}\rightarrow \mathcal {C}^{D_\mathrm {a}\mathord \partial \Delta [m]}_\mathrm {R}\text {.} \end{aligned}$$</span></div></div><p>We form pullbacks (the front and back squares of the cube) </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-av"><figure><div class="c-article-section__figure-content" id="Figav"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figav_HTML.gif?as=webp"><img aria-describedby="Figav" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figav_HTML.gif" alt="figure av" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-av-desc"></div></div></figure></div><p> Observe that <span class="mathjax-tex">$\mathcal {C}^{\widetilde{D}(K \star \Delta [m])}$</span> and <span class="mathjax-tex">$\mathcal {C}^{\widetilde{D}(K \star \mathord \partial \Delta [m])}$</span> are just atomic notations for the pullbacks above, i.e. <span class="mathjax-tex">$\widetilde{D}(K \star \Delta [m])$</span> and <span class="mathjax-tex">$\widetilde{D}(K \star \mathord \partial \Delta [m])$</span> are <i>not</i> homotopical categories for general <i>K</i>, although they will be interpreted as such when <i>K</i> is a simplex.</p> <h3 class="c-article__sub-heading" id="FPar73">Lemma 4.5</h3> <p>The induced functor <span class="mathjax-tex">$P_K :\mathcal {C}^{\widetilde{D}(K \star \Delta [m])}_\mathrm {R}\rightarrow \mathcal {C}^{\widetilde{D}(K \star \mathord \partial \Delta [m])}_\mathrm {R}$</span> is an acyclic fibration for every <span class="mathjax-tex">$\kappa $</span>-small simplicial set <i>K</i>.</p> <h3 class="c-article__sub-heading" id="FPar74">Proof</h3> <p>First, we verify that <span class="mathjax-tex">$P_K$</span> is a fibration. The categories <span class="mathjax-tex">$\mathcal {C}^{\widetilde{D}(K \star \Delta [m])}_\mathrm {R}$</span> and <span class="mathjax-tex">$\mathcal {C}^{\widetilde{D}(K \star \mathord \partial \Delta [m])}_\mathrm {R}$</span> are full subcategories of <span class="mathjax-tex">$\mathcal {C}^{D(K \star \Delta [m])}_\mathrm {R}$</span> and <span class="mathjax-tex">$\mathcal {C}^{D(K \star \mathord \partial \Delta [m])}_\mathrm {R}$</span> respectively. They are both closed under taking weakly equivalent objects. Hence <span class="mathjax-tex">$P_K$</span> inherits the desired lifting properties from the fibration <span class="mathjax-tex">$\mathcal {C}^{D(K \star \Delta [m])}_\mathrm {R}\twoheadrightarrow \mathcal {C}^{D(K \star \mathord \partial \Delta [m])}_\mathrm {R}$</span>.</p> <p>For the rest of the argument it will suffice to check that <span class="mathjax-tex">$P_K$</span> is a weak equivalence when <i>K</i> is empty or a simplex and that this property is preserved under coproducts, pushouts along monomorphisms and colimits of sequences of monomorphisms.</p> <p>When <i>K</i> is empty then the top square of the cube above happens to be a pullback and hence <span class="mathjax-tex">$P_\varnothing $</span> is an acyclic fibration by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar71">4.4</a>.</p> <p>For <span class="mathjax-tex">$K = \Delta [k]$</span> we will check that <span class="mathjax-tex">$P_{\Delta [k]}$</span> coincides with</p><div id="Equ17" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \mathcal {C}^{D[k+\widetilde{1}+m]}_\mathrm {R}\rightarrow \mathcal {C}^{D\Lambda ^{[k]}[k+\widetilde{1}+m]}_\mathrm {R}\end{aligned}$$</span></div></div><p>and the conclusion will follow from Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar59">3.14</a>. It is enough to verify that a homotopical Reedy cofibrant diagram <span class="mathjax-tex">$X :D[k+1+m] \rightarrow \mathcal {C}$</span> is homotopical with respect to <span class="mathjax-tex">$D[k+\widetilde{1}+m]$</span> if and only if the induced morphism</p><div id="Equ18" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\text {colim}}_{D[k]} X \rightarrow {\text {colim}}_{D[k+1]} X \end{aligned}$$</span></div></div><p>is a weak equivalence. This follows from Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar65">4.1</a>. The same argument works with <span class="mathjax-tex">$\Lambda ^{[k]}[k+1+m]$</span> in place of <span class="mathjax-tex">$[k+1+m]$</span>, since <span class="mathjax-tex">$\Delta [k+1]$</span> is contained in <span class="mathjax-tex">$\Lambda ^{[k]}[k+1+m]$</span> for <span class="mathjax-tex">$m > 0$</span>.</p> <p>If </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-aw"><figure><div class="c-article-section__figure-content" id="Figaw"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figaw_HTML.gif?as=webp"><img aria-describedby="Figaw" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figaw_HTML.gif" alt="figure aw" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-aw-desc"></div></div></figure></div> <p>is a pushout square of simplicial sets such that the conclusion holds for <i>A</i>, <i>B</i> and <i>K</i>, then there is a pullback square of cofibration categories </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-ax"><figure><div class="c-article-section__figure-content" id="Figax"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figax_HTML.gif?as=webp"><img aria-describedby="Figax" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figax_HTML.gif" alt="figure ax" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-ax-desc"></div></div></figure></div><p> An analogous pullback square with <i>D</i> in the place of <span class="mathjax-tex">$\widetilde{D}$</span> results directly from Corollary <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar63">3.16</a> (1). This implies the existence of the one above since pullbacks commute with pullbacks. Similarly, there is an analogous pullback square with <span class="mathjax-tex">$\mathord \partial \Delta [m]$</span> in place of <span class="mathjax-tex">$\Delta [m]$</span>. Hence the conclusion for <i>L</i> follows from the Gluing Lemma.</p> <p>The last two cases depend on the fact that <span class="mathjax-tex">$\mathsf {CofCat}_\kappa $</span> is a complete fibration category as follows from [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Szumiło, K.: Homotopy theory of cofibration categories. Homol. Homotop. Appl (to appear)" href="/article/10.1007/s40062-016-0139-x#ref-CR21" id="ref-link-section-d81634044e34176">21</a>, Theorem 2.9].</p> <p>If <i>K</i> is a colimit of a sequence of monomorphisms <span class="mathjax-tex">$K_0 \hookrightarrow K_1 \hookrightarrow K_2 \hookrightarrow \cdots $</span>, then <span class="mathjax-tex">$\mathcal {C}^{\widetilde{D}(K \star \Delta [m])}_\mathrm {R}$</span> is the limit of the tower of fibrations</p><div id="Equ19" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \cdots \twoheadrightarrow \mathcal {C}^{\widetilde{D}(K_2 \star \Delta [m])}_\mathrm {R}\twoheadrightarrow \mathcal {C}^{\widetilde{D}(K_1 \star \Delta [m])}_\mathrm {R}\twoheadrightarrow \mathcal {C}^{\widetilde{D}(K_0 \star \Delta [m])}_\mathrm {R}\end{aligned}$$</span></div></div><p>and analogously for <span class="mathjax-tex">$\mathcal {C}^{\widetilde{D}(K \star \mathord \partial \Delta [m])}_\mathrm {R}$</span> [this follows from Corollary <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar63">3.16</a> (2), similarly to the previous case]. Therefore, if <span class="mathjax-tex">$P_{K_i}$</span> is a weak equivalence for all <i>i</i>, then so is <span class="mathjax-tex">$P_K$</span>.</p> <p>The case of coproducts is handled similarly except that <span class="mathjax-tex">${-}\star \Delta [m]$</span> does not preserve coproducts but carries them to wide pushouts. Hence <span class="mathjax-tex">$\mathcal {C}^{\widetilde{D}((\coprod _i K_i) \star \Delta [m])}_\mathrm {R}$</span> is the wide pullback</p><div id="Equ44" class="c-article-equation"><div class="c-article-equation__content"><img src="//media.springernature.com/full/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Equ44_HTML.gif" class="u-display-block" alt=""></div></div><p>The conclusion follows since the wide pullback functor is an exact functor of fibration categories. <span class="mathjax-tex">$\square $</span> </p> <p>We are ready to characterize colimits in <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> in terms of homotopy colimits in <span class="mathjax-tex">$\mathcal {C}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar75">Theorem 4.6</h3> <p>Let <span class="mathjax-tex">$\mathcal {C}$</span> be a <span class="mathjax-tex">$\kappa $</span>-cocomplete cofibration category, <i>K</i> a <span class="mathjax-tex">$\kappa $</span>-small simplicial set and <span class="mathjax-tex">$S :K^\rhd \rightarrow \mathrm{N}_\mathrm{f}\mathcal {C}$</span>. Then <i>S</i> is universal as a cone under <i>S</i>|<i>K</i> if and only if the induced morphism</p><div id="Equ20" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\text {colim}}_{DK} S \rightarrow {\text {colim}}_{D(K^\rhd )} S \end{aligned}$$</span></div></div><p>is a weak equivalence (with <i>S</i> seen as a homotopical Reedy cofibrant diagram <span class="mathjax-tex">$D(K^\rhd ) \rightarrow \mathcal {C}$</span> by Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar33">2.6</a>). Such a cone exists under every diagram <span class="mathjax-tex">$K \rightarrow \mathrm{N}_\mathrm{f}\mathcal {C}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar76">Proof</h3> <p>If the morphism above is a weak equivalence let <span class="mathjax-tex">$U :K \star \mathord \partial \Delta [m] \rightarrow \mathrm{N}_\mathrm{f}\mathcal {C}$</span> extend <i>S</i>. The functor <span class="mathjax-tex">$\mathcal {C}^{\widetilde{D}(K \star \Delta [m])}_\mathrm {R}\rightarrow \mathcal {C}^{\widetilde{D}(K \star \mathord \partial \Delta [m])}_\mathrm {R}$</span> is an acyclic fibration by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar73">4.5</a> and thus the corresponding homotopical Reedy cofibrant diagram <span class="mathjax-tex">$\widetilde{D}(K \star \mathord \partial \Delta [m]) \rightarrow \mathcal {C}$</span> prolongs to <span class="mathjax-tex">$\widetilde{D}(K \star \Delta [m]) \rightarrow \mathcal {C}$</span>. Hence <i>S</i> is universal.</p> <p>Conversely, let <i>S</i> be universal. Define <span class="mathjax-tex">$T :D(K^\rhd ) \rightarrow \mathcal {C}$</span> as in Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar67">4.2</a> where we take <i>f</i> to be the identity of <span class="mathjax-tex">${\text {colim}}_{D K} S$</span>. Then the induced morphism <span class="mathjax-tex">${\text {colim}}_{DK} T \rightarrow {\text {colim}}_{D(K^\rhd )} T$</span> is a weak equivalence and so <i>T</i> is universal by the argument above (which proves the existence statement). Therefore <i>S</i> and <i>T</i> are equivalent and hence there exists a homotopical Reedy cofibrant diagram <span class="mathjax-tex">$W :D(K \star {E[1]}) \rightarrow \mathcal {C}$</span> which restricts to <i>S</i> on <span class="mathjax-tex">$D(K \star \{ 0 \})$</span> and to <i>T</i> on <span class="mathjax-tex">$D(K \star \{ 1 \})$</span>, see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="Joyal, A.: Quasi-categories and Kan complexes. J. Pure Appl. Algebra 175(1–3), 207–222 (2002). doi:
 10.1016/S0022-4049(02)00135-4
 
 . Special volume celebrating the 70th birthday of Professor Max Kelly. MR1935979" href="/article/10.1007/s40062-016-0139-x#ref-CR9" id="ref-link-section-d81634044e35788">9</a>, Proposition 4.4]. In the diagram </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-ay"><figure><div class="c-article-section__figure-content" id="Figay"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figay_HTML.gif?as=webp"><img aria-describedby="Figay" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figay_HTML.gif" alt="figure ay" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-ay-desc"></div></div></figure></div> <p>both bottom horizontal morphisms and the top right one are weak equivalences by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar65">4.1</a> and so is the right vertical one since the homotopical structure of <span class="mathjax-tex">$D{E[1]}$</span> is the maximal one. It follows that <span class="mathjax-tex">${\text {colim}}_{DK} S \rightarrow {\text {colim}}_{D(K^\rhd )} S$</span> is also a weak equivalence. <span class="mathjax-tex">$\square $</span> </p> </div></div></section><section data-title="Cocompleteness: the finite case"><div class="c-article-section" id="Sec6-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec6"><span class="c-article-section__title-number">6 </span>Cocompleteness: the finite case</h2><div class="c-article-section__content" id="Sec6-content"><p>In this section we will prove that <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> is finitely cocomplete for any cofibration category. The arguments of the previous section do not directly apply to this case since they heavily use the existence of colimits of Reedy cofibrant diagrams over categories of the form <i>DK</i>. Unfortunately, <i>DK</i> is infinite even when <i>K</i> is a finite (non-empty) simplicial set. In order to address this problem, we will filter the category <i>DK</i> by finite subcategories</p><div id="Equ21" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} D^{(0)}K \hookrightarrow D^{(1)}K \hookrightarrow D^{(2)}K \hookrightarrow \cdots \end{aligned}$$</span></div></div><p>and instead of using a colimit of a Reedy cofibrant diagram <span class="mathjax-tex">$X :DK \rightarrow \mathcal {C}$</span> we will consider the resulting sequence of finite colimits</p><div id="Equ22" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\text {colim}}_{D^{(0)}K} X \rightarrowtail {\text {colim}}_{D^{(1)}K} X \rightarrowtail {\text {colim}}_{D^{(2)}K} X \rightarrowtail \cdots \end{aligned}$$</span></div></div><p>If <i>X</i> is homotopical this sequence stabilizes in the sense that from some point on (depending on <i>K</i>) all morphisms are weak equivalences and this stable value is a homotopy colimit of <i>X</i>. However, there is no universal bound on when such a sequence stabilizes when <i>K</i> varies and hence we are forced to think of that entire sequence as a homotopy colimit of <i>X</i>. It turns out that the proofs of the previous section will work if we carefully substitute such sequences for actual colimits over categories <i>DK</i>. The difficult part is constructing such filtrations with all the desired naturality and homotopy invariance which is the main purpose of this section.</p><p>Let <i>J</i> be a homotopical category and <i>A</i> a set of objects of <i>DJ</i>, we denote the sieve generated by <i>A</i> in <i>DJ</i> by <span class="mathjax-tex">$D^{A}J$</span>. Moreover, when <span class="mathjax-tex">$J = [m]$</span> (possibly with some non-trivial homotopical structure) we will write objects of <i>D</i>[<i>m</i>] as non-decreasing sequences of elements of [<i>m</i>] often using abbreviations like <span class="mathjax-tex">$0^k 1^l$</span> to denote the sequence of <i>k</i> 0s followed by <i>l</i> 1s.</p><p>The category <i>D</i>[0] can be seen as the category of non-degenerate simplices of a simplicial set <i>S</i> with exactly one non-degenerate simplex in each dimension. As it turns out, the skeleton <span class="mathjax-tex">${\text {Sk}}^k S$</span> is weakly contractible for <i>k</i> even but weakly equivalent to the sphere <span class="mathjax-tex">$\Delta [k] / \mathord \partial \Delta [k]$</span> for <i>k</i> odd.</p><p>This suggests that the filtration of <i>D</i>[0] by sieves generated by even-dimensional simplices of <i>S</i> should be well-behaved homotopically. We verify that this is the case in the next two lemmas and later generalize it to <i>DK</i> for arbitrary finite simplicial sets <i>K</i>.</p> <h3 class="c-article__sub-heading" id="FPar77">Lemma 5.1</h3> <p>For each <i>k</i> the functor <span class="mathjax-tex">$t :D^{0^k 1}\widehat{[1]} \rightarrow [0]$</span> is a homotopy equivalence of homotopical categories.</p> <h3 class="c-article__sub-heading" id="FPar78">Proof</h3> <p>Represent objects of <span class="mathjax-tex">$D^{0^k 1}\widehat{[1]}$</span> as binary sequences and let <span class="mathjax-tex">$j :[0] \rightarrow D^{0^k 1}\widehat{[1]}$</span> classify the object 1. Next, define <span class="mathjax-tex">$s :D^{0^k 1}\widehat{[1]} \rightarrow D^{0^k 1}\widehat{[1]}$</span> by appending a trailing 1 to each sequence that doesn’t have one. Then there are natural weak equivalences </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-az"><figure><div class="c-article-section__figure-content" id="Figaz"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figaz_HTML.gif?as=webp"><img aria-describedby="Figaz" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figaz_HTML.gif" alt="figure az" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-az-desc"></div></div></figure></div><p> Moreover, we have <span class="mathjax-tex">$t j = {\mathrm {id}}_{[0]}$</span> which finishes the proof. <span class="mathjax-tex">$\square $</span> </p> <p>The images of the composite functors</p><div id="Equ23" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\text {Sd}}[k] \hookrightarrow D[k] \rightarrow D[0] \quad \text { and }\quad {\text {Sd}}\mathord \partial \Delta [k+1] \hookrightarrow D \mathord \partial \Delta [k+1] \rightarrow D[0] \end{aligned}$$</span></div></div><p>are both <span class="mathjax-tex">$D^{0^{k+1}}[0]$</span>. In the next lemma we consider the resulting functors</p><div id="Equ24" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} t :{\text {Sd}}[k] \rightarrow D^{0^{k+1}}[0] \quad \text { and }\quad t :{\text {Sd}}\mathord \partial \Delta [k+1] \rightarrow D^{0^{k+1}}[0] \text {.} \end{aligned}$$</span></div></div> <h3 class="c-article__sub-heading" id="FPar79">Lemma 5.2</h3> <p>Let <span class="mathjax-tex">$k \ge 0$</span> and let <span class="mathjax-tex">$\mathcal {C}$</span> be a cofibration category. If <span class="mathjax-tex">$X :D^{0^{k+1}}[0] \rightarrow \mathcal {C}$</span> is a homotopical Reedy cofibrant diagram, then</p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(1)</span> <p>the induced morphism </p><div id="Equ25" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\text {colim}}_{{\text {Sd}}\Delta [k]} t^* X \rightarrow {\text {colim}}_{D^{0^{k+1}}[0]} X \end{aligned}$$</span></div></div><p> is a weak equivalence when <i>k</i> is even,</p> </li> <li> <span class="u-custom-list-number">(2)</span> <p>the induced morphism </p><div id="Equ26" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\text {colim}}_{{\text {Sd}}\mathord \partial \Delta [k+1]} t^* X \rightarrow {\text {colim}}_{D^{0^{k+1}}[0]} X \end{aligned}$$</span></div></div><p> is a weak equivalence when <i>k</i> is odd.</p> </li> </ol> <h3 class="c-article__sub-heading" id="FPar80">Proof</h3> <p>We prove both statements by an alternating induction with respect to <i>k</i>.</p> <p>The functor <span class="mathjax-tex">${\text {Sd}}[0] \rightarrow D^{0}[0]$</span> is an isomorphism, so condition (1) holds for <span class="mathjax-tex">$k = 0$</span>.</p> <p>Next, we assume that condition (2) holds for a given odd <i>k</i> and prove that condition (1) holds for <span class="mathjax-tex">$k+1$</span>. The category <span class="mathjax-tex">${\text {Sd}}\mathord \partial \Delta [k+1]$</span> is nothing but the latching category of <span class="mathjax-tex">$D^{0^{k+2}}[0]$</span> at <span class="mathjax-tex">$0^{k+2}$</span> and hence the inductive construction of the colimit of <i>X</i> yields a pushout square </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-ba"><figure><div class="c-article-section__figure-content" id="Figba"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figba_HTML.gif?as=webp"><img aria-describedby="Figba" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figba_HTML.gif" alt="figure ba" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-ba-desc"></div></div></figure></div> <p>where the top morphism is a weak equivalence by the inductive hypothesis. Since the left vertical morphism is a cofibration, it follows by the Gluing Lemma that the bottom morphism is also a weak equivalence.</p> <p>Finally, we assume that condition (1) holds for a given even <i>k</i> and prove that condition (2) holds for <span class="mathjax-tex">$k+1$</span>. We have the following diagram of homotopical direct categories </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-bb"><figure><div class="c-article-section__figure-content" id="Figbb"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbb_HTML.gif?as=webp"><img aria-describedby="Figbb" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbb_HTML.gif" alt="figure bb" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-bb-desc"></div></div></figure></div> <p>where the indicated maps are sieves, the top left and bottom right squares are pushouts and all functors respect Reedy cofibrant diagrams by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar13">1.10</a> (the functor on the very left is induced by <span class="mathjax-tex">$0^{k+2} 1 :[k+2] \rightarrow [1]$</span>). Hence there is an induced diagram in <span class="mathjax-tex">$\mathcal {C}$</span> </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-bc"><figure><div class="c-article-section__figure-content" id="Figbc"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbc_HTML.gif?as=webp"><img aria-describedby="Figbc" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbc_HTML.gif" alt="figure bc" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-bc-desc"></div></div></figure></div> <p>where the indicated maps are cofibrations by [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Răadulescu-Banu, A.: Cofibrations in Homotopy Theory (2006). 
 http://arxiv.org/abs/math/0610009v4
 
 
 " href="/article/10.1007/s40062-016-0139-x#ref-CR14" id="ref-link-section-d81634044e38006">14</a>, Theorem 9.4.1 (1a)] and the top left and bottom right squares are pushouts (here, we use the fact that a colimit of a diagram whose indexing category is a colimit of a diagram of categories can be computed as an iterated colimit).</p> <p>Thus the proof will be completed when we verify that both morphisms</p><div id="Equ27" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\text {colim}}_{{\text {Sd}}\Lambda ^{\widehat{k+2}}\hat{[k+2]}} t^* X&\rightarrow {\text {colim}}_{D^{0^{k+1} 1}\widehat{[1]}} t^* X \\ {\text {colim}}_{D^{0^{k+1}}[0]} t^* X&\rightarrow {\text {colim}}_{D^{0^{k+1} 1}\widehat{[1]}} X \end{aligned}$$</span></div></div><p>are weak equivalences. For the former we use [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title="Kapulkin, K., Szumiło, K.: Quasicategories of frames of cofibration categories. Appl. Categ. Struct., 1–25 (2016)" href="/article/10.1007/s40062-016-0139-x#ref-CR13" id="ref-link-section-d81634044e38290">13</a>, Lemma 3.17] and Lemmas <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar77">5.1</a> and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar57">3.13</a>. For the latter we use [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title="Kapulkin, K., Szumiło, K.: Quasicategories of frames of cofibration categories. Appl. Categ. Struct., 1–25 (2016)" href="/article/10.1007/s40062-016-0139-x#ref-CR13" id="ref-link-section-d81634044e38299">13</a>, Lemma 3.17], Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar77">5.1</a> and the inductive assumption. <span class="mathjax-tex">$\square $</span> </p> <p>In the next two lemmas we generalize the filtration of <i>D</i>[0] to <i>D</i>[<i>m</i>] for all <span class="mathjax-tex">$m \ge 0$</span>.</p> <h3 class="c-article__sub-heading" id="FPar81">Lemma 5.3</h3> <p>Let <span class="mathjax-tex">$\mathcal {C}$</span> be a cofibration category. Assume that every fiber of <span class="mathjax-tex">$\varphi :[k] \rightarrow [m]$</span> has an odd number of elements and let <span class="mathjax-tex">$X :D^\varphi [m] \rightarrow \mathcal {C}$</span> be a homotopical Reedy cofibrant diagram. Then <span class="mathjax-tex">$X_\varphi \rightarrow {\text {colim}}X$</span> is a weak equivalence.</p> <h3 class="c-article__sub-heading" id="FPar82">Proof</h3> <p>We proceed by induction with respect to <i>m</i> (simultaneously for all <span class="mathjax-tex">$\mathcal {C}$</span> and <i>X</i>). For <span class="mathjax-tex">$m = 0$</span> the conclusion follows by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar79">5.2</a>.</p> <p>If <span class="mathjax-tex">$m > 0$</span>, we will prolong <i>X</i> to the augmented sieve <span class="mathjax-tex">$D_\mathrm {a}^\varphi [m]$</span> by setting the missing value to an initial object of <span class="mathjax-tex">$\mathcal {C}$</span> which does not change the colimit. If the fiber of <span class="mathjax-tex">$\varphi $</span> over <i>m</i> has <span class="mathjax-tex">$k+1$</span> elements for some even <i>k</i>, then <span class="mathjax-tex">$D_\mathrm {a}^\varphi [m] \cong D_\mathrm {a}^{\varphi '}[m-1] \times D_\mathrm {a}^{0^{k+1}}[0]$</span> (here, <span class="mathjax-tex">$\varphi '$</span> is the restriction of <span class="mathjax-tex">$\varphi $</span> to <span class="mathjax-tex">$\varphi ^{-1}[m-1]$</span>). We apply Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar79">5.2</a> in the category <span class="mathjax-tex">$\mathcal {C}^{D_\mathrm {a}^{\varphi '}[m-1]}_\mathrm {R}$</span> to the corresponding diagram <span class="mathjax-tex">$\widetilde{X} :D_\mathrm {a}^{0^{k+1}}[0] \rightarrow \mathcal {C}^{D_\mathrm {a}^{\varphi '}[m-1]}_\mathrm {R}$</span> (this diagram is indeed Reedy cofibrant by the proof of Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar19">1.13</a>). This way, we obtain a weak equivalence <span class="mathjax-tex">$\widetilde{X}_k \rightarrow {\text {colim}}_{D_\mathrm {a}^{0^{k+1}}[0]} \widetilde{X}$</span> and hence by the inductive assumption the composite</p><div id="Equ28" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} X_\varphi = \widetilde{X}_{k,\varphi '} \rightarrow {\text {colim}}_{D_\mathrm {a}^{\varphi '}[m-1]} \widetilde{X}_k \rightarrow {\text {colim}}_{D_\mathrm {a}^{\varphi '}[m-1]} {\text {colim}}_{D_\mathrm {a}^{0^{k+1}}[0]} \widetilde{X} \cong {\text {colim}}X \end{aligned}$$</span></div></div><p>is also a weak equivalence. <span class="mathjax-tex">$\square $</span> </p> <p>For each <span class="mathjax-tex">$k, m \ge 0$</span> we define sets <span class="mathjax-tex">$A_{k,m}$</span> and <span class="mathjax-tex">$B_{k,m}$</span> of objects of <i>D</i>[<i>m</i>]. We proceed by induction with respect to <i>m</i>. First, we set <span class="mathjax-tex">$A_{k,0} = B_{k,0} = \{ [2k] \rightarrow [0]\}$</span>. For <span class="mathjax-tex">$m > 0$</span> we set</p><div id="Equ29" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} B_{k,m}&= \{ \varphi :[2k-m] \rightarrow [m] \mid \text {each fiber of } \varphi \text { has an odd number of elements} \} \\ A_{k,m}&= B_{k,m} \cup \bigcup _{i \in [m]} \delta _i A_{k,m-1} \text {.} \end{aligned}$$</span></div></div><p>We set <span class="mathjax-tex">$D^{(k)}[m] = D^{A_{k,m}}[m]$</span>. In particular, we have <span class="mathjax-tex">$D^{(k)}[0] = D^{[2k]}[0]$</span>, i.e. <span class="mathjax-tex">$D^{(k)} [m]$</span> generalizes the filtration of <i>D</i>[0] by sieves corresponding to even-dimensional skeleta of <i>S</i> as discussed on p. 33. There <i>S</i> was defined as a simplicial set with exactly one non-degenerate simplex in each dimension. Similarly, there is a simplicial set <span class="mathjax-tex">$S_m$</span> whose non-degenerate <i>p</i>-simplices correspond to all simplicial operators <span class="mathjax-tex">$[p] \rightarrow [m]$</span> (with degenerate ones adjoined freely). Then the sieve generated by <span class="mathjax-tex">$B_{k,m}$</span> corresponds to the (2<i>k</i>)-skeleton of <span class="mathjax-tex">$S_m$</span>. However, these sieves do not match exactly when <i>m</i> varies and the definition of <span class="mathjax-tex">$A_{k,m}$</span> corrects that.</p> <h3 class="c-article__sub-heading" id="FPar83">Lemma 5.4</h3> <p>For every simplicial operator <span class="mathjax-tex">$\chi :[m] \rightarrow [n]$</span> and <span class="mathjax-tex">$k \ge 0$</span> we have an inclusion <span class="mathjax-tex">$\chi D^{(k)}[m] \subseteq D^{(k)}[n]$</span>.</p> <h3 class="c-article__sub-heading" id="FPar84">Proof</h3> <p>It suffices to verify the statement when <span class="mathjax-tex">$\chi $</span> is an elementary face or degeneracy operator. For the elementary face operators it follows directly from the definition. Hence assume that <span class="mathjax-tex">$\chi = \sigma _j$</span> for some <span class="mathjax-tex">$j \in [n]$</span>. We will check that <span class="mathjax-tex">$\sigma _j A_{k,n+1} \subseteq D^{(k)}[n]$</span> by induction with respect to <i>n</i>.</p> <p>If <span class="mathjax-tex">$\varphi :[2k-n-1] \rightarrow [n+1]$</span> has all fibers of odd cardinality, then the same holds for <span class="mathjax-tex">$\sigma _j \varphi $</span> except at the fiber over <i>j</i>. Then <span class="mathjax-tex">$\sigma _j \varphi $</span> is in the sieve generated by <span class="mathjax-tex">$\varphi ' :[2k-n] \rightarrow [n]$</span> obtained by adding one extra element to the fiber of <span class="mathjax-tex">$\sigma _j \varphi $</span> over <i>j</i> (so that <span class="mathjax-tex">$\varphi ' \in A_{k,n}$</span>).</p> <p>If <span class="mathjax-tex">$\psi \in A_{k,n}$</span>, then <span class="mathjax-tex">$\sigma _j \delta _i \psi $</span> is either equal to <span class="mathjax-tex">$\psi $</span> or is of the form <span class="mathjax-tex">$\delta _{i'} \sigma _{j'} \psi $</span>. In the first case the conclusion holds trivially, in the second one it follows by the inductive hypothesis. <span class="mathjax-tex">$\square $</span> </p> <p>Now, we can generalize the filtration of <i>D</i>[<i>m</i>] to <i>DK</i> for arbitrary finite <i>K</i>. Let <span class="mathjax-tex">$x \in K_m$</span> and <span class="mathjax-tex">$k \ge 0$</span>. We define a sieve <span class="mathjax-tex">$D^{(k)}K$</span> in <i>DK</i> as follows. Write <span class="mathjax-tex">$x = x^\sharp x^\flat $</span> with <span class="mathjax-tex">$x^\sharp $</span> non-degenerate and <span class="mathjax-tex">$x^\flat $</span> a degeneracy operator (this can be done in exactly one way by the Eilenberg–Zilber Lemma, see e.g. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="Gabriel, P., Zisman, M.: Calculus of Fractions and Homotopy Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 35. Springer-Verlag New York Inc, New York (1967)" href="/article/10.1007/s40062-016-0139-x#ref-CR7" id="ref-link-section-d81634044e41243">7</a>, Section II.3]). Define <i>x</i> to be an element of <span class="mathjax-tex">$D^{(k)}K$</span> if <span class="mathjax-tex">$x^\flat \in D^{(k)}[n]$</span> (where <i>n</i> is the dimension of <span class="mathjax-tex">$x^\sharp $</span>). It follows from Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar83">5.4</a> that this definition coincides with the previous one when <i>K</i> is a simplex.</p> <h3 class="c-article__sub-heading" id="FPar85">Lemma 5.5</h3> <p>Every simplicial map <span class="mathjax-tex">$f :K \rightarrow L$</span> carries <span class="mathjax-tex">$D^{(k)}K$</span> to <span class="mathjax-tex">$D^{(k)}L$</span> for all <span class="mathjax-tex">$k \ge 0$</span>.</p> <h3 class="c-article__sub-heading" id="FPar86">Proof</h3> <p>Let <span class="mathjax-tex">$x \in D^{(k)}K$</span>. Then we have a diagram of simplicial sets </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-bd"><figure><div class="c-article-section__figure-content" id="Figbd"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbd_HTML.gif?as=webp"><img aria-describedby="Figbd" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbd_HTML.gif" alt="figure bd" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-bd-desc"></div></div></figure></div> <p>and by definition <span class="mathjax-tex">$x^\flat \in D^{(k)}[n]$</span>. Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar83">5.4</a> implies that <span class="mathjax-tex">$(fx)^\flat \in D^{(k)}[n']$</span> so that <span class="mathjax-tex">$fx \in D^{(k)}L$</span>. <span class="mathjax-tex">$\square $</span> </p> <h3 class="c-article__sub-heading" id="FPar87">Lemma 5.6</h3> <p>For all <span class="mathjax-tex">$k \ge m$</span>, a cofibration category <span class="mathjax-tex">$\mathcal {C}$</span> and a homotopical Reedy cofibrant diagram <span class="mathjax-tex">$X :D^{(k)}[m] \rightarrow \mathcal {C}$</span> the morphism <span class="mathjax-tex">$X_{[m]} \rightarrow {\text {colim}}_{D^{(k)}[m]} X$</span> is a weak equivalence.</p> <h3 class="c-article__sub-heading" id="FPar88">Proof</h3> <p>First, we will check that the morphism <span class="mathjax-tex">$X_{[m]} \rightarrow D^{B_{k,m}}[m]$</span> is a weak equivalence. Indeed, let <i>P</i> be the subposet of <span class="mathjax-tex">$\mathbb {N}^{m+1}$</span> consisting of tuples <span class="mathjax-tex">$x = (x_0, \ldots , x_m)$</span> such that each <span class="mathjax-tex">$x_i$</span> is odd and <span class="mathjax-tex">$x_0 + \cdots + x_m \le 2k - m + 1$</span>. Let <span class="mathjax-tex">$\varphi _x$</span> be the unique object of <i>D</i>[<i>m</i>] whose fiber over each <span class="mathjax-tex">$i \in [m]$</span> has cardinality <span class="mathjax-tex">$x_i$</span>. Then we have <span class="mathjax-tex">$D^{B_{k,m}}[m] = {\text {colim}}_{x \in P} D^{\varphi _x}[m]$</span> since a colimit of a diagram whose indexing category is a colimit of a diagram of categories can be computed as an iterated colimit.</p> <p>It follows from Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar81">5.3</a> that for each <span class="mathjax-tex">$x \in P$</span> the morphism <span class="mathjax-tex">$X_{[m]} \rightarrow {\text {colim}}_{D^{\varphi _x}[m]} X$</span> is a weak equivalence. The sequence <span class="mathjax-tex">$(1,\ldots ,1)$</span> is the bottom element of <i>P</i>, hence if we consider <i>P</i> as a homotopical poset with all maps as weak equivalences, then <span class="mathjax-tex">$\{(1,\ldots ,1)\} \rightarrow P$</span> is a homotopy equivalence. It follows by [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title="Kapulkin, K., Szumiło, K.: Quasicategories of frames of cofibration categories. Appl. Categ. Struct., 1–25 (2016)" href="/article/10.1007/s40062-016-0139-x#ref-CR13" id="ref-link-section-d81634044e42584">13</a>, Lemma 3.17] that <span class="mathjax-tex">$X_{[m]} \rightarrow D^{B_{k,m}}[m]$</span> is a weak equivalence.</p> <p>We are ready to prove the lemma by induction with respect to <i>m</i>. If <span class="mathjax-tex">$m = 0$</span>, then the conclusion is a special case of Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar81">5.3</a>. For <span class="mathjax-tex">$m > 0$</span>, we consider diagrams <span class="mathjax-tex">$Y^{(k-1)}, Y^{(k)} :{\text {Sd}}\mathord \partial \Delta [m] \rightarrow \mathcal {C}$</span> defined as</p><div id="Equ30" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} Y^{(k-1)}_\varphi = {\text {colim}}_{D^{\varphi A_{k-1,n}}[m]} X \quad \text { and }\quad Y^{(k)}_\varphi = {\text {colim}}_{D^{\varphi A_{k,n}}[m]} X \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$\varphi :[n] \hookrightarrow [m]$</span>. The resulting morphism <span class="mathjax-tex">$Y^{(k-1)} \rightarrow Y^{(k)}$</span> is a Reedy cofibration by [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Răadulescu-Banu, A.: Cofibrations in Homotopy Theory (2006). 
 http://arxiv.org/abs/math/0610009v4
 
 
 " href="/article/10.1007/s40062-016-0139-x#ref-CR14" id="ref-link-section-d81634044e43094">14</a>, Theorem 9.4.1 (1a)]. It is also a weak equivalence by the inductive hypothesis since <span class="mathjax-tex">$D^{\varphi A_{k-1,n}}[m] \cong D^{A_{k-1,n}}[n]$</span> and <span class="mathjax-tex">$D^{\varphi A_{k,n}}[m] \cong D^{A_{k,n}}[n]$</span>. Therefore, the induced morphism <span class="mathjax-tex">${\text {colim}}Y^{(k-1)} \rightarrow {\text {colim}}Y^{(k)}$</span> is an acyclic cofibration. This morphism is also isomorphic to <span class="mathjax-tex">${\text {colim}}_{D^{(k-1)}\mathord \partial \Delta [m]} X \rightarrow {\text {colim}}_{D^{(k)}\mathord \partial \Delta [m]} X$</span> since</p><div id="Equ31" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} D^{(k-1)}\mathord \partial \Delta [m]&= {\text {colim}}_{\varphi \in {\text {Sd}}\mathord \partial \Delta [m]} D^{\varphi A_{k-1,n}}[m] \\ \text {and }\quad D^{(k)}\mathord \partial \Delta [m]&= {\text {colim}}_{\varphi \in {\text {Sd}}\mathord \partial \Delta [m]} D^{\varphi A_{k,n}}[m] \text {.} \end{aligned}$$</span></div></div><p>Finally, we observe that <span class="mathjax-tex">$D^{B_{k,m}}[m] \cap D^{(k)}\mathord \partial \Delta [m] = D^{(k-1)}\mathord \partial \Delta [m]$</span>. This yields pushout square </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-be"><figure><div class="c-article-section__figure-content" id="Figbe"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbe_HTML.gif?as=webp"><img aria-describedby="Figbe" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbe_HTML.gif" alt="figure be" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-be-desc"></div></div></figure></div> <p>which along with the first part of the proof yields the final conclusion. <span class="mathjax-tex">$\square $</span> </p> <h3 class="c-article__sub-heading" id="FPar89">Lemma 5.7</h3> <p>For each <i>k</i> the functor <span class="mathjax-tex">$D^{(k)} :\mathsf {sSet}\rightarrow \mathsf {Cat}$</span> (i.e. when we disregard the homotopical structures of <span class="mathjax-tex">$D^{(k)}K$</span>s) preserves colimits.</p> <h3 class="c-article__sub-heading" id="FPar90">Proof</h3> <p>If <i>K</i> is any simplicial set, then <span class="mathjax-tex">$D^{(k)}$</span> preserves the colimit of its simplices by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar31">2.5</a> and the definition of <span class="mathjax-tex">$D^{(k)}K$</span>. Hence for every small category <i>J</i> we have the following sequence of isomorphisms natural in both <i>K</i> and <i>J</i>.</p><div id="Equ32" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \mathsf {Cat}(D^{(k)}K,J)&\cong \mathsf {Cat}(D^{(k)}{\text {colim}}_{\Delta [m] \rightarrow K} \Delta [m], J) \\&\cong \lim _{\Delta [m] \rightarrow K} \mathsf {Cat}(D^{(k)}[m], J) \\&\cong \lim _{\Delta [m] \rightarrow K} \mathsf {sSet}(\Delta [m], \mathsf {Cat}(D^{(k)}[{-}], J)) \\&\cong \mathsf {sSet}(K, \mathsf {Cat}(D^{(k)}[{-}], J)) \end{aligned}$$</span></div></div><p>It follows that <span class="mathjax-tex">$J \mapsto \mathsf {Cat}(D^{(k)}[{-}], J)$</span> is a right adjoint of <span class="mathjax-tex">$D^{(k)}$</span> and the conclusion follows. <span class="mathjax-tex">$\square $</span> </p> <p>Finally, we are ready to start translating the results of Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s40062-016-0139-x#Sec5">4</a> to the case of <span class="mathjax-tex">$\kappa = \aleph _0$</span>. The following is a counterpart to Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar65">4.1</a>.</p> <h3 class="c-article__sub-heading" id="FPar91">Lemma 5.8</h3> <p>Let <span class="mathjax-tex">$\mathcal {C}$</span> be a cofibration category and <i>K</i> a finite simplicial set. For every homotopical Reedy cofibrant diagram <span class="mathjax-tex">$X :D(K \star \Delta [m]) \rightarrow \mathcal {C}$</span> and all <span class="mathjax-tex">$k \ge \dim K + 1 + m$</span>, the induced morphism</p><div id="Equ33" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} X_{[m]} \rightarrow {\text {colim}}_{D^{(k)}(K \star \Delta [m])} X \end{aligned}$$</span></div></div><p>is a weak equivalence.</p> <h3 class="c-article__sub-heading" id="FPar92">Proof</h3> <p>The proof is analogous to the proof of Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar65">4.1</a> using Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar89">5.7</a> in the place of Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar31">2.5</a>. <span class="mathjax-tex">$\square $</span> </p> <p>For a cofibration category <span class="mathjax-tex">$\mathcal {C}$</span> we introduce a new cofibration category <span class="mathjax-tex">$\mathcal {C}^{{\widetilde{\mathbb {N}}}}_\mathrm {R}$</span> (here, <span class="mathjax-tex">${\widetilde{\mathbb {N}}}$</span> does not refer to any homotopical structure on <span class="mathjax-tex">$\mathbb {N}$</span>, <span class="mathjax-tex">$\mathcal {C}^{{\widetilde{\mathbb {N}}}}_\mathrm {R}$</span> should be seen as an atomic notation). Its objects are Reedy cofibrant diagrams <span class="mathjax-tex">$X :\mathbb {N}\rightarrow \mathcal {C}$</span> (i.e. sequences of cofibrations in <span class="mathjax-tex">$\mathcal {C}$</span>) that are <i>eventually (homotopically) constant</i>, i.e. such that there is a number <i>k</i> such that for all <span class="mathjax-tex">$l \ge k$</span> the morphism <span class="mathjax-tex">$X_k \rightarrow X_l$</span> is a weak equivalence. A morphism <span class="mathjax-tex">$f :X \rightarrow Y$</span> of such diagrams is called an <i>eventual weak equivalence</i> if there is <i>k</i> such that for all <span class="mathjax-tex">$l \ge k$</span> the morphism <span class="mathjax-tex">$f_l$</span> is a weak equivalence in <span class="mathjax-tex">$\mathcal {C}$</span>. This cofibration category is designed as an enlargement of the cofibration category <span class="mathjax-tex">$\mathcal {C}^{{\widehat{\mathbb {N}}}}_\mathrm {R}$</span> of <i>(homotopically) constant</i> sequences. It is necessary since sequences arising as colimits over filtrations <span class="mathjax-tex">$D^{(-)}K$</span> are only eventually constant.</p> <h3 class="c-article__sub-heading" id="FPar93">Lemma 5.9</h3> <p>If <span class="mathjax-tex">$\mathcal {C}$</span> is a cofibration category, then the category <span class="mathjax-tex">$\mathcal {C}^{{\widetilde{\mathbb {N}}}}_\mathrm {R}$</span> with Reedy cofibrations and eventual weak equivalences is also a cofibration category. Moreover, the inclusion <span class="mathjax-tex">$\mathcal {C}^{{\widehat{\mathbb {N}}}}_\mathrm {R}\hookrightarrow \mathcal {C}^{{\widetilde{\mathbb {N}}}}_\mathrm {R}$</span> is a weak equivalence.</p> <h3 class="c-article__sub-heading" id="FPar94">Proof</h3> <p>The construction of the cofibration category <span class="mathjax-tex">$\mathcal {C}^{{\widetilde{\mathbb {N}}}}_\mathrm {R}$</span> is a straightforward modification of the construction of <span class="mathjax-tex">$\mathcal {C}^\mathbb {N}_\mathrm {R}$</span>, see e.g. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Răadulescu-Banu, A.: Cofibrations in Homotopy Theory (2006). 
 http://arxiv.org/abs/math/0610009v4
 
 
 " href="/article/10.1007/s40062-016-0139-x#ref-CR14" id="ref-link-section-d81634044e45583">14</a>, Theorem 9.3.5 (1)].</p> <p>We will verify the approximation properties [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Szumiło, K.: Homotopy theory of cofibration categories. Homol. Homotop. Appl (to appear)" href="/article/10.1007/s40062-016-0139-x#ref-CR21" id="ref-link-section-d81634044e45589">21</a>, Proposition 2.2]. By 2-out-of-3 a morphism between homotopically constant sequences is a levelwise weak equivalence if and only if it is an eventual weak equivalence. Hence (App1) holds.</p> <p>Next, let <span class="mathjax-tex">$X \rightarrow Y$</span> be a morphism with <i>X</i> homotopically constant and <i>Y</i> eventually constant. Assume that <i>Y</i> is homotopically constant from degree <i>k</i> on. Let <span class="mathjax-tex">$\widetilde{Y}$</span> be <i>Y</i> shifted down by <i>k</i>. Then <span class="mathjax-tex">$\widetilde{Y}$</span> is homotopically constant and iterated structure morphisms of <i>Y</i> yield a morphism <span class="mathjax-tex">$Y \rightarrow \widetilde{Y}$</span> which is an eventual weak equivalence (starting from <i>k</i>). This yields a commutative square </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-bf"><figure><div class="c-article-section__figure-content" id="Figbf"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbf_HTML.gif?as=webp"><img aria-describedby="Figbf" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbf_HTML.gif" alt="figure bf" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-bf-desc"></div></div></figure></div> <p>which proves (App2). <span class="mathjax-tex">$\square $</span> </p> <p>We define a functor <span class="mathjax-tex">$|{-}| :DK \rightarrow \mathbb {N}$</span> by sending <span class="mathjax-tex">$x \in DK$</span> to the smallest <span class="mathjax-tex">$k \in \mathbb {N}$</span> such that <span class="mathjax-tex">$x \in D^{(k)}K$</span>. We call |<i>x</i>| the <i>filtration degree</i> of <i>x</i>. Here, we do not consider any particular homotopical structure on <span class="mathjax-tex">$\mathbb {N}$</span> so <span class="mathjax-tex">$|{-}|$</span> is not a homotopical functor. We will be interested in the left Kan extension of a homotopical Reedy cofibrant diagram <span class="mathjax-tex">$X :DK \rightarrow \mathcal {C}$</span> along <span class="mathjax-tex">$|{-}|$</span>. It can be computed as</p><div id="Equ34" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} ({\text {Lan}}_{|{-}|} X)_k = {\text {colim}}_{D^{(k)}K} X \text {.} \end{aligned}$$</span></div></div><p>We will denote <span class="mathjax-tex">$({\text {Lan}}_{|{-}|} X)_k$</span> by <span class="mathjax-tex">$\Phi ^{(k)} X$</span> and when <i>k</i> varies <span class="mathjax-tex">$\Phi ^{({-})} X$</span> will stand for the resulting sequence <span class="mathjax-tex">$\mathbb {N}\rightarrow \mathcal {C}$</span>.</p><p>Just as colimits can be defined in terms of cones, left Kan extensions can be defined in terms of certain generalized cones. We describe such cones for Kan extensions along <span class="mathjax-tex">$|{-}|$</span>. Let <span class="mathjax-tex">$DK \star _{|{-}|} \mathbb {N}$</span> denote the <i>cograph</i> (or <i>collage</i>) of <span class="mathjax-tex">$|{-}|$</span> defined as the category whose set of objects is the disjoint union of the sets of objects of <i>DK</i> and <span class="mathjax-tex">$\mathbb {N}$</span> and</p><div id="Equ35" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} (DK \star _{|{-}|} \mathbb {N}) (x, y) = {\left\{ \begin{array}{ll} DK(x, y) &{} \quad \text {when } x, y \in DK \text {,} \\ \mathbb {N}(x, y) &{} \quad \text {when } x, y \in \mathbb {N}\text {,} \\ \mathbb {N}(|x|, y) &{} \quad \text {when } x \in DK \text { and } y \in \mathbb {N}\text {,} \\ \varnothing &{} \quad \text {otherwise.} \end{array}\right. } \end{aligned}$$</span></div></div><p>Alternatively, <span class="mathjax-tex">$DK \star _{|{-}|} \mathbb {N}$</span> can be constructed as the pushout square </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-bg"><figure><div class="c-article-section__figure-content" id="Figbg"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbg_HTML.gif?as=webp"><img aria-describedby="Figbg" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbg_HTML.gif" alt="figure bg" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-bg-desc"></div></div></figure></div><p> The left Kan extension of <span class="mathjax-tex">$X :DK \rightarrow \mathcal {C}$</span> along <span class="mathjax-tex">$|{-}|$</span> is nothing but an initial extension of <i>X</i> to <span class="mathjax-tex">$DK \star _{|{-}|} \mathbb {N}$</span> so that morphisms <span class="mathjax-tex">$\Phi ^{({-})} X \rightarrow Y$</span> in <span class="mathjax-tex">$\mathcal {C}^\mathbb {N}$</span> correspond to diagrams on <span class="mathjax-tex">$DK \star _{|{-}|} \mathbb {N}$</span> restricting to <i>X</i> and <i>Y</i> on <i>DK</i> and <span class="mathjax-tex">$\mathbb {N}$</span> respectively. Such an extension of <i>X</i> is a family of cones under the restrictions of <i>X</i> to all <span class="mathjax-tex">$D^{(k)}K$</span>s. We will compare them to extensions to <span class="mathjax-tex">$D(K^\rhd )$</span> using a functor <span class="mathjax-tex">$p_K :D(K^\rhd ) \rightarrow DK \star _{|{-}|} \mathbb {N}$</span> defined as follows. Write an object of <span class="mathjax-tex">$D(K^\rhd )$</span> as <span class="mathjax-tex">$x \star \varphi $</span> with <span class="mathjax-tex">$x \in D^a K$</span> and <span class="mathjax-tex">$\varphi \in D^a [0]$</span> and set</p><div id="Equ36" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} p_K(x \star \varphi ) = {\left\{ \begin{array}{ll} |x \star \varphi | &{}\quad \text {when } \varphi \in D[0] \text {,} \\ x &{}\quad \text {otherwise.} \end{array}\right. } \end{aligned}$$</span></div></div><p>This allows us to state and prove a version of Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar67">4.2</a> for finitely cocomplete cofibration categories.</p> <h3 class="c-article__sub-heading" id="FPar95">Lemma 5.10</h3> <p>Let <span class="mathjax-tex">$\mathcal {C}$</span> be a cofibration category, <i>K</i> a finite simplicial set and <span class="mathjax-tex">$X :DK \rightarrow \mathcal {C}$</span> a homotopical Reedy cofibrant diagram. Consider a morphism <span class="mathjax-tex">$f :\Phi ^{({-})} X \rightarrow Y$</span> and the corresponding cone <span class="mathjax-tex">$\widetilde{T} :DK \star _{|{-}|} \mathbb {N}\rightarrow \mathcal {C}$</span>. If <i>T</i> is any Reedy cofibrant replacement of <span class="mathjax-tex">$p_K^* \widetilde{T}$</span> relative to <i>DK</i> (i.e. <i>T</i> is Reedy cofibrant and comes with a weak equivalence <span class="mathjax-tex">$T \mathop {\rightarrow }\limits ^{\sim }p_K^* \widetilde{T}$</span> that is an identity over <i>DK</i>, such a replacement exists by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar11">1.9</a>), then <i>f</i> factors as</p><div id="Equ37" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Phi ^{({-})} X \rightarrow \Phi ^{({-})} T \mathop {\rightarrow }\limits ^{\sim }Y \end{aligned}$$</span></div></div><p>where the latter morphism is an eventual weak equivalence (starting at <span class="mathjax-tex">$\dim K + 1$</span>).</p> <h3 class="c-article__sub-heading" id="FPar96">Proof</h3> <p>To verify that the above composite agrees with <i>f</i> it suffices to check that at each level <i>k</i> it agrees upon precomposition with <span class="mathjax-tex">$X_x \rightarrow \Phi ^{(k)} X$</span> for all <span class="mathjax-tex">$x \in D^{(k)}K$</span>. That’s indeed the case since <span class="mathjax-tex">$T|DK = X$</span>.</p> <p>It remains to check that the latter morphism is an eventual weak equivalence. For <span class="mathjax-tex">$i \ge \dim K + 1$</span> in the diagram</p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-bh"><figure><div class="c-article-section__figure-content" id="Figbh"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbh_HTML.gif?as=webp"><img aria-describedby="Figbh" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbh_HTML.gif" alt="figure bh" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-bh-desc"></div></div></figure></div> <p>the left morphism is a weak equivalence by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar91">5.8</a> and so is the right one since <i>T</i> is a cofibrant replacement of <span class="mathjax-tex">$p_K^* \widetilde{T}$</span>. The bottom morphism is a weak equivalence by the homotopical structure of <span class="mathjax-tex">$D(K^\rhd )$</span> and therefore so is the the top one. <span class="mathjax-tex">$\square $</span> </p> <p>For every <span class="mathjax-tex">$m \ge 0$</span> each object of <span class="mathjax-tex">$D(K \star \Delta [m])$</span> can be uniquely written as <span class="mathjax-tex">$x \star \varphi $</span> with <span class="mathjax-tex">$x \in D^a K$</span> and <span class="mathjax-tex">$\varphi \in D^a[m]$</span>. This yields a functor <span class="mathjax-tex">$r_K :D(K \star \Delta [m]) \rightarrow D^a[m]$</span> sending <span class="mathjax-tex">$x \star \varphi $</span> to <span class="mathjax-tex">$\varphi $</span> and to which we can associate the “filtered” left Kan extension functor</p><div id="Equ38" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\text {Lan}}_{r_K}^{\mathrm {filt}} :\mathcal {C}^{D(K \star \Delta [m])}_\mathrm {R}\rightarrow (\mathcal {C}^{{\widetilde{\mathbb {N}}}}_\mathrm {R})^{D^a[m]}_\mathrm {R}\end{aligned}$$</span></div></div><p>defined as <span class="mathjax-tex">$({\text {Lan}}_{r_K}^{\mathrm {filt}} X)_\varphi = \Phi ^{({-})} \varphi ^* X$</span> for <span class="mathjax-tex">$\varphi \in D^a[m]$</span>. The functor <span class="mathjax-tex">${\text {Lan}}_{r_K}^{\mathrm {filt}}$</span> is exact by [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Răadulescu-Banu, A.: Cofibrations in Homotopy Theory (2006). 
 http://arxiv.org/abs/math/0610009v4
 
 
 " href="/article/10.1007/s40062-016-0139-x#ref-CR14" id="ref-link-section-d81634044e48752">14</a>, Theorem 9.4.3(1)]. Similarly we have</p><div id="Equ39" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\text {Lan}}_{s_K}^{\mathrm {filt}} :(\mathcal {C}^{{\widetilde{\mathbb {N}}}}_\mathrm {R})^{D(K \star \mathord \partial \Delta [m])}_\mathrm {R}\rightarrow (\mathcal {C}^{{\widetilde{\mathbb {N}}}}_\mathrm {R})^{D^a\mathord \partial \Delta [m]}_\mathrm {R}\end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$s_K :D(K \star \mathord \partial \Delta [m]) \rightarrow D^a\mathord \partial \Delta [m]$</span> is a functor defined in the same way as <span class="mathjax-tex">$r_K$</span>. We form pullbacks (the front and back squares of the cube) </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-bi"><figure><div class="c-article-section__figure-content" id="Figbi"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbi_HTML.gif?as=webp"><img aria-describedby="Figbi" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs40062-016-0139-x/MediaObjects/40062_2016_139_Figbi_HTML.gif" alt="figure bi" loading="lazy"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-bi-desc"></div></div></figure></div><p> Observe that <span class="mathjax-tex">$\mathcal {C}^{\widetilde{D}(K \star \Delta [m])}$</span> and <span class="mathjax-tex">$\mathcal {C}^{\widetilde{D}(K \star \mathord \partial \Delta [m])}$</span> are just atomic notations for the pullbacks above, i.e. <span class="mathjax-tex">$\widetilde{D}(K \star \Delta [m])$</span> and <span class="mathjax-tex">$\widetilde{D}(K \star \mathord \partial \Delta [m])$</span> are <i>not</i> homotopical categories for general <i>K</i>, although they will be interpreted as such when <i>K</i> is a simplex.</p><p>The following is a finite variant of Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar73">4.5</a>.</p> <h3 class="c-article__sub-heading" id="FPar97">Lemma 5.11</h3> <p>The functor <span class="mathjax-tex">$P_K :\mathcal {C}^{\widetilde{D}(K \star \Delta [m])}_\mathrm {R}\rightarrow \mathcal {C}^{\widetilde{D}(K \star \mathord \partial \Delta [m])}_\mathrm {R}$</span> is an acyclic fibration for every finite simplicial set <i>K</i>.</p> <h3 class="c-article__sub-heading" id="FPar98">Proof</h3> <p>The proof is virtually identical to the proof of Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar73">4.5</a> except that now we do not consider the cases of coproducts and colimits of sequences of monomorphisms and we use Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar91">5.8</a> in the place of Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar65">4.1</a>. <span class="mathjax-tex">$\square $</span> </p> <p>Finally, we can characterize colimits in <span class="mathjax-tex">$\mathrm{N}_\mathrm{f}\mathcal {C}$</span> in terms of homotopy colimits in <span class="mathjax-tex">$\mathcal {C}$</span> in a manner similar to Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar75">4.6</a>.</p> <h3 class="c-article__sub-heading" id="FPar99">Theorem 5.12</h3> <p>Let <span class="mathjax-tex">$\mathcal {C}$</span> be cofibration category, <i>K</i> a finite simplicial set. A cone <span class="mathjax-tex">$S :K^\rhd \rightarrow \mathrm{N}_\mathrm{f}\mathcal {C}$</span> is universal if and only if the induced morphism</p><div id="Equ40" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Phi ^{({-})}(S|K) \rightarrow \Phi ^{({-})} S \end{aligned}$$</span></div></div><p>is an eventual weak equivalence (where <i>S</i> is seen as a homotopical Reedy cofibrant diagram <span class="mathjax-tex">$D(K^\rhd ) \rightarrow \mathcal {C}$</span> by Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar33">2.6</a>). Such a cone exists under every diagram <span class="mathjax-tex">$K \rightarrow \mathrm{N}_\mathrm{f}\mathcal {C}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar100">Proof</h3> <p>The proof is almost identical to the proof of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar75">4.6</a> except that we use Lemmas <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar91">5.8</a>, <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar95">5.10</a> and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar97">5.11</a> in the place of Lemmas <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar65">4.1</a>, <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar67">4.2</a> and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s40062-016-0139-x#FPar73">4.5</a> respectively. <span class="mathjax-tex">$\square $</span> </p> </div></div></section> </div> <section data-title="Notes"><div class="c-article-section" id="notes-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="notes">Notes</h2><div class="c-article-section__content" id="notes-content"><ol class="c-article-footnote c-article-footnote--listed"><li class="c-article-footnote--listed__item" id="Fn1" data-counter="1."><div class="c-article-footnote--listed__content"><p>This result differs from its counterpart for model categories since it uses presimplicial sets (a.k.a. <span class="mathjax-tex">$\Delta $</span>-sets or semisimplicial sets) as models of homotopy types. Presimplicial sets are less well-behaved than simplicial sets, but their homotopy theory is equivalent to that of simplicial sets, see e.g. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Rourke, C.P., Sanderson, B.J.: 
 
 
 
 $$\triangle $$
 
 
 ▵
 
 
 -sets. I. Homotopy theory. Quart. J. Math. Oxford Ser. 22(2), 321–338 (1971)" href="/article/10.1007/s40062-016-0139-x#ref-CR16" id="ref-link-section-d81634044e12724">16</a>, Proposition 2.1].</p></div></li></ol></div></div></section><div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Bib1">References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ol class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="1."><p class="c-article-references__text" id="ref-CR1">Avigad, J., Kapulkin, K., Lumsdaine, P.L.: Homotopy limits in type theory. Math. Struct. Comput. Sci. <b>25</b>(5), 1040–1070 (2015)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1017/S0960129514000498" data-track-item_id="10.1017/S0960129514000498" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1017%2FS0960129514000498" aria-label="Article reference 1" data-doi="10.1017/S0960129514000498">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3340534" aria-label="MathSciNet reference 1">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1362.18004" aria-label="MATH reference 1">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 1" href="http://scholar.google.com/scholar_lookup?&title=Homotopy%20limits%20in%20type%20theory&journal=Math.%20Struct.%20Comput.%20Sci.&doi=10.1017%2FS0960129514000498&volume=25&issue=5&pages=1040-1070&publication_year=2015&author=Avigad%2CJ&author=Kapulkin%2CK&author=Lumsdaine%2CPL"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="2."><p class="c-article-references__text" id="ref-CR2">Borceux, F.: Handbook of categorical algebra. 1. Encyclopedia of Mathematics and its Applications, vol. 50, Cambridge University Press, Cambridge. Basic category theory (1994)</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="3."><p class="c-article-references__text" id="ref-CR3">Brown, K.S.: Abstract homotopy theory and generalized sheaf cohomology. Trans. Am. Math. Soc. <b>186</b>, 419–458 (1973)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1090/S0002-9947-1973-0341469-9" data-track-item_id="10.1090/S0002-9947-1973-0341469-9" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1090%2FS0002-9947-1973-0341469-9" aria-label="Article reference 3" data-doi="10.1090/S0002-9947-1973-0341469-9">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=341469" aria-label="MathSciNet reference 3">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0245.55007" aria-label="MATH reference 3">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 3" href="http://scholar.google.com/scholar_lookup?&title=Abstract%20homotopy%20theory%20and%20generalized%20sheaf%20cohomology&journal=Trans.%20Am.%20Math.%20Soc.&doi=10.1090%2FS0002-9947-1973-0341469-9&volume=186&pages=419-458&publication_year=1973&author=Brown%2CKS"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="4."><p class="c-article-references__text" id="ref-CR4">Cordier, J.M.: Sur la notion de diagramme homotopiquement cohérent. Cahiers Topologie Géom. Différentielle <b>23</b>(1), 93–112 (1982) (French). Third Colloquium on Categories, Part VI (Amiens, 1980)</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="5."><p class="c-article-references__text" id="ref-CR5">Dwyer, W.G., Kan, D.M.: Simplicial localizations of categories. J. Pure Appl. Algebra <b>17</b>(3), 267–284 (1980)</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="6."><p class="c-article-references__text" id="ref-CR6">Dwyer, W.G., Kan, D.M.: Calculating simplicial localizations. J. Pure Appl. Algebra <b>18</b>(1), 17–35 (1980)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1016/0022-4049(80)90113-9" data-track-item_id="10.1016/0022-4049(80)90113-9" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1016%2F0022-4049%2880%2990113-9" aria-label="Article reference 6" data-doi="10.1016/0022-4049(80)90113-9">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=578563" aria-label="MathSciNet reference 6">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0485.18013" aria-label="MATH reference 6">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 6" href="http://scholar.google.com/scholar_lookup?&title=Calculating%20simplicial%20localizations&journal=J.%20Pure%20Appl.%20Algebra&doi=10.1016%2F0022-4049%2880%2990113-9&volume=18&issue=1&pages=17-35&publication_year=1980&author=Dwyer%2CWG&author=Kan%2CDM"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="7."><p class="c-article-references__text" id="ref-CR7">Gabriel, P., Zisman, M.: Calculus of Fractions and Homotopy Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 35. Springer-Verlag New York Inc, New York (1967)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="noopener" data-track-label="10.1007/978-3-642-85844-4" data-track-item_id="10.1007/978-3-642-85844-4" data-track-value="book reference" data-track-action="book reference" href="https://link.springer.com/doi/10.1007/978-3-642-85844-4" aria-label="Book reference 7" data-doi="10.1007/978-3-642-85844-4">Book</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 7" href="http://scholar.google.com/scholar_lookup?&title=Calculus%20of%20Fractions%20and%20Homotopy%20Theory%2C%20Ergebnisse%20der%20Mathematik%20und%20ihrer%20Grenzgebiete&doi=10.1007%2F978-3-642-85844-4&publication_year=1967&author=Gabriel%2CP&author=Zisman%2CM"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="8."><p class="c-article-references__text" id="ref-CR8">Hovey, M.: Model categories, Mathematical Surveys and Monographs, vol. 63. American Mathematical Society, Providence, RI (1999)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 8" href="http://scholar.google.com/scholar_lookup?&title=Model%20categories%2C%20Mathematical%20Surveys%20and%20Monographs&publication_year=1999&author=Hovey%2CM"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="9."><p class="c-article-references__text" id="ref-CR9">Joyal, A.: Quasi-categories and Kan complexes. J. Pure Appl. Algebra <b>175</b>(1–3), 207–222 (2002). doi:<a href="https://doi.org/10.1016/S0022-4049(02)00135-4" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="10.1016/S0022-4049(02)00135-4">10.1016/S0022-4049(02)00135-4</a>. Special volume celebrating the 70th birthday of Professor Max Kelly. MR1935979</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="10."><p class="c-article-references__text" id="ref-CR10">Joyal, A.: The Theory of Quasi-Categories and its Applications. Quadern 45, Vol. II, Centre de Recerca Matemàtica Barcelona (2008)</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="11."><p class="c-article-references__text" id="ref-CR11">Joyal, A., Tierney, M.: Quasi-categories vs Segal spaces. Categories in algebra, geometry and mathematical physics, Contemp. Math., vol. 431, pp. 277–326. Amer. Math. Soc., Providence, RI (2007)</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="12."><p class="c-article-references__text" id="ref-CR12">Kapulkin, K.: Locally Cartesian Closed Quasicategories From Type Theory. <a href="http://arxiv.org/abs/1507.02648" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="http://arxiv.org/abs/1507.02648">arXiv:1507.02648</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="13."><p class="c-article-references__text" id="ref-CR13">Kapulkin, K., Szumiło, K.: Quasicategories of frames of cofibration categories. Appl. Categ. Struct., 1–25 (2016)</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="14."><p class="c-article-references__text" id="ref-CR14">Răadulescu-Banu, A.: Cofibrations in Homotopy Theory (2006). <a href="http://arxiv.org/abs/math/0610009v4" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="http://arxiv.org/abs/math/0610009v4">http://arxiv.org/abs/math/0610009v4</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="15."><p class="c-article-references__text" id="ref-CR15">Rezk, C.: A model for the homotopy theory of homotopy theory. Trans. Amer. Math. Soc. <b>353</b>(3), 973–1007 (2001) (electronic)</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="16."><p class="c-article-references__text" id="ref-CR16">Rourke, C.P., Sanderson, B.J.: <span class="mathjax-tex">$\triangle $</span>-sets. I. Homotopy theory. Quart. J. Math. Oxford Ser. <b>22</b>(2), 321–338 (1971)</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="17."><p class="c-article-references__text" id="ref-CR17">Riehl, E., Verity, D.: The theory and practice of Reedy categories. Theory Appl. Categ. <b>29</b>, 256–301 (2014)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3217884" aria-label="MathSciNet reference 17">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1302.55014" aria-label="MATH reference 17">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 17" href="http://scholar.google.com/scholar_lookup?&title=The%20theory%20and%20practice%20of%20Reedy%20categories&journal=Theory%20Appl.%20Categ.&volume=29&pages=256-301&publication_year=2014&author=Riehl%2CE&author=Verity%2CD"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="18."><p class="c-article-references__text" id="ref-CR18">Schwede, S.: The p-order of topological triangulated categories. J. Topol. <b>6</b>(4), 868–914 (2013)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1112/jtopol/jtt018" data-track-item_id="10.1112/jtopol/jtt018" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1112%2Fjtopol%2Fjtt018" aria-label="Article reference 18" data-doi="10.1112/jtopol/jtt018">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3145143" aria-label="MathSciNet reference 18">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1294.18008" aria-label="MATH reference 18">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 18" href="http://scholar.google.com/scholar_lookup?&title=The%20p-order%20of%20topological%20triangulated%20categories&journal=J.%20Topol.&doi=10.1112%2Fjtopol%2Fjtt018&volume=6&issue=4&pages=868-914&publication_year=2013&author=Schwede%2CS"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="19."><p class="c-article-references__text" id="ref-CR19">Szumiło, K.: Two Models for the Homotopy Theory of Cocomplete Homotopy Theories. Ph.D. Thesis, Rheinische Friedrich-Wilhelms-Universität Bonn (2014). <a href="http://hss.ulb.uni-bonn.de/2014/3692/3692.htm" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="http://hss.ulb.uni-bonn.de/2014/3692/3692.htm">http://hss.ulb.uni-bonn.de/2014/3692/3692.htm</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="20."><p class="c-article-references__text" id="ref-CR20">Szumiło, K.: Two Models for the Homotopy Theory of Cocomplete Homotopy Theories (2014). <a href="http://arxiv.org/abs/1411.0303" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="http://arxiv.org/abs/1411.0303">arXiv:1411.0303</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="21."><p class="c-article-references__text" id="ref-CR21">Szumiło, K.: Homotopy theory of cofibration categories. Homol. Homotop. Appl (to appear)</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="22."><p class="c-article-references__text" id="ref-CR22">Szumiło, K.: Homotopy theory of cocomplete quasicategories. Algebraic Geom Topol (to appear)</p></li></ol><p class="c-article-references__download u-hide-print"><a data-track="click" data-track-action="download citation references" data-track-label="link" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1007/s40062-016-0139-x?format=refman&flavour=references">Download references<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p></div></div></div></section></div><section data-title="Acknowledgments"><div class="c-article-section" id="Ack1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Ack1">Acknowledgments</h2><div class="c-article-section__content" id="Ack1-content"><p>This paper is based on a part of my thesis [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Szumiło, K.: Two Models for the Homotopy Theory of Cocomplete Homotopy Theories. Ph.D. Thesis, Rheinische Friedrich-Wilhelms-Universität Bonn (2014). 
 http://hss.ulb.uni-bonn.de/2014/3692/3692.htm
 
 
 " href="/article/10.1007/s40062-016-0139-x#ref-CR19" id="ref-link-section-d81634044e49885">19</a>] which was written while I was a doctoral student in <i>Bonn International Graduate School in Mathematics</i> and, more specifically, <i>Graduiertenkolleg 1150 “Homotopy and Cohomology”</i> and <i>International Max Planck Research School for Moduli Spaces</i>. I want to thank everyone involved for creating an excellent working environment. I want to thank Clark Barwick, Bill Dwyer, André Joyal, Chris Kapulkin, Lennart Meier, Thomas Nikolaus, Chris Schommer-Pries, Peter Teichner and Marek Zawadowski for conversations on various topics which were very beneficial to my research. I am especially grateful to Viktoriya Ozornova and Irakli Patchkoria for reading an early draft of my thesis. Their feedback helped me make many improvements and avoid numerous errors. Above all, I want to express my gratitude to my supervisor Stefan Schwede whose expertise was always invaluable and without whose support this thesis could not have been written.</p></div></div></section><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="author-information">Author information</h2><div class="c-article-section__content" id="author-information-content"><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff1"><p class="c-article-author-affiliation__address">Department of Mathematics, University of Western Ontario, 1151 Richmond Street, London, ON, N6A 3K7, Canada</p><p class="c-article-author-affiliation__authors-list">Karol Szumiło</p></li></ol><div class="u-js-hide u-hide-print" data-test="author-info"><span class="c-article__sub-heading">Authors</span><ol class="c-article-authors-search u-list-reset"><li id="auth-Karol-Szumi_o-Aff1"><span class="c-article-authors-search__title u-h3 js-search-name">Karol Szumiło</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?dc.creator=Karol%20Szumi%C5%82o" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text">You can also search for this author in <span class="c-article-identifiers"><a class="c-article-identifiers__item" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Karol%20Szumi%C5%82o" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="http://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Karol%20Szumi%C5%82o%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li></ol></div><h3 class="c-article__sub-heading" id="corresponding-author">Corresponding author</h3><p id="corresponding-author-list">Correspondence to <a id="corresp-c1" href="mailto:kszumilo@uwo.ca">Karol Szumiło</a>.</p></div></div></section><section data-title="Additional information"><div class="c-article-section" id="additional-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="additional-information">Additional information</h2><div class="c-article-section__content" id="additional-information-content"><p>Communicated by Steve Wilson.</p></div></div></section><section data-title="Rights and permissions"><div class="c-article-section" id="rightslink-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="rightslink">Rights and permissions</h2><div class="c-article-section__content" id="rightslink-content"><p class="c-article-rights"><a data-track="click" data-track-action="view rights and permissions" data-track-label="link" href="https://s100.copyright.com/AppDispatchServlet?title=Frames%20in%20cofibration%20categories&author=Karol%20Szumi%C5%82o&contentID=10.1007%2Fs40062-016-0139-x&copyright=Tbilisi%20Centre%20for%20Mathematical%20Sciences&publication=2193-8407&publicationDate=2016-07-13&publisherName=SpringerNature&orderBeanReset=true">Reprints and permissions</a></p></div></div></section><section aria-labelledby="article-info" data-title="About this article"><div class="c-article-section" id="article-info-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="article-info">About this article</h2><div class="c-article-section__content" id="article-info-content"><div class="c-bibliographic-information"><div class="u-hide-print c-bibliographic-information__column c-bibliographic-information__column--border"><a data-crossmark="10.1007/s40062-016-0139-x" target="_blank" rel="noopener" href="https://crossmark.crossref.org/dialog/?doi=10.1007/s40062-016-0139-x" data-track="click" data-track-action="Click Crossmark" data-track-label="link" data-test="crossmark"><img loading="lazy" width="57" height="81" alt="Check for updates. Verify currency and authenticity via CrossMark" src="data:image/svg+xml;base64,<svg height="81" width="57" xmlns="http://www.w3.org/2000/svg"><g fill="none" fill-rule="evenodd"><path d="m17.35 35.45 21.3-14.2v-17.03h-21.3" fill="#989898"/><path d="m38.65 35.45-21.3-14.2v-17.03h21.3" fill="#747474"/><path d="m28 .5c-12.98 0-23.5 10.52-23.5 23.5s10.52 23.5 23.5 23.5 23.5-10.52 23.5-23.5c0-6.23-2.48-12.21-6.88-16.62-4.41-4.4-10.39-6.88-16.62-6.88zm0 41.25c-9.8 0-17.75-7.95-17.75-17.75s7.95-17.75 17.75-17.75 17.75 7.95 17.75 17.75c0 4.71-1.87 9.22-5.2 12.55s-7.84 5.2-12.55 5.2z" fill="#535353"/><path d="m41 36c-5.81 6.23-15.23 7.45-22.43 2.9-7.21-4.55-10.16-13.57-7.03-21.5l-4.92-3.11c-4.95 10.7-1.19 23.42 8.78 29.71 9.97 6.3 23.07 4.22 30.6-4.86z" fill="#9c9c9c"/><path d="m.2 58.45c0-.75.11-1.42.33-2.01s.52-1.09.91-1.5c.38-.41.83-.73 1.34-.94.51-.22 1.06-.32 1.65-.32.56 0 1.06.11 1.51.35.44.23.81.5 1.1.81l-.91 1.01c-.24-.24-.49-.42-.75-.56-.27-.13-.58-.2-.93-.2-.39 0-.73.08-1.05.23-.31.16-.58.37-.81.66-.23.28-.41.63-.53 1.04-.13.41-.19.88-.19 1.39 0 1.04.23 1.86.68 2.46.45.59 1.06.88 1.84.88.41 0 .77-.07 1.07-.23s.59-.39.85-.68l.91 1c-.38.43-.8.76-1.28.99-.47.22-1 .34-1.58.34-.59 0-1.13-.1-1.64-.31-.5-.2-.94-.51-1.31-.91-.38-.4-.67-.9-.88-1.48-.22-.59-.33-1.26-.33-2.02zm8.4-5.33h1.61v2.54l-.05 1.33c.29-.27.61-.51.96-.72s.76-.31 1.24-.31c.73 0 1.27.23 1.61.71.33.47.5 1.14.5 2.02v4.31h-1.61v-4.1c0-.57-.08-.97-.25-1.21-.17-.23-.45-.35-.83-.35-.3 0-.56.08-.79.22-.23.15-.49.36-.78.64v4.8h-1.61zm7.37 6.45c0-.56.09-1.06.26-1.51.18-.45.42-.83.71-1.14.29-.3.63-.54 1.01-.71.39-.17.78-.25 1.18-.25.47 0 .88.08 1.23.24.36.16.65.38.89.67s.42.63.54 1.03c.12.41.18.84.18 1.32 0 .32-.02.57-.07.76h-4.36c.07.62.29 1.1.65 1.44.36.33.82.5 1.38.5.29 0 .57-.04.83-.13s.51-.21.76-.37l.55 1.01c-.33.21-.69.39-1.09.53-.41.14-.83.21-1.26.21-.48 0-.92-.08-1.34-.25-.41-.16-.76-.4-1.07-.7-.31-.31-.55-.69-.72-1.13-.18-.44-.26-.95-.26-1.52zm4.6-.62c0-.55-.11-.98-.34-1.28-.23-.31-.58-.47-1.06-.47-.41 0-.77.15-1.07.45-.31.29-.5.73-.58 1.3zm2.5.62c0-.57.09-1.08.28-1.53.18-.44.43-.82.75-1.13s.69-.54 1.1-.71c.42-.16.85-.24 1.31-.24.45 0 .84.08 1.17.23s.61.34.85.57l-.77 1.02c-.19-.16-.38-.28-.56-.37-.19-.09-.39-.14-.61-.14-.56 0-1.01.21-1.35.63-.35.41-.52.97-.52 1.67 0 .69.17 1.24.51 1.66.34.41.78.62 1.32.62.28 0 .54-.06.78-.17.24-.12.45-.26.64-.42l.67 1.03c-.33.29-.69.51-1.08.65-.39.15-.78.23-1.18.23-.46 0-.9-.08-1.31-.24-.4-.16-.75-.39-1.05-.7s-.53-.69-.7-1.13c-.17-.45-.25-.96-.25-1.53zm6.91-6.45h1.58v6.17h.05l2.54-3.16h1.77l-2.35 2.8 2.59 4.07h-1.75l-1.77-2.98-1.08 1.23v1.75h-1.58zm13.69 1.27c-.25-.11-.5-.17-.75-.17-.58 0-.87.39-.87 1.16v.75h1.34v1.27h-1.34v5.6h-1.61v-5.6h-.92v-1.2l.92-.07v-.72c0-.35.04-.68.13-.98.08-.31.21-.57.4-.79s.42-.39.71-.51c.28-.12.63-.18 1.04-.18.24 0 .48.02.69.07.22.05.41.1.57.17zm.48 5.18c0-.57.09-1.08.27-1.53.17-.44.41-.82.72-1.13.3-.31.65-.54 1.04-.71.39-.16.8-.24 1.23-.24s.84.08 1.24.24c.4.17.74.4 1.04.71s.54.69.72 1.13c.19.45.28.96.28 1.53s-.09 1.08-.28 1.53c-.18.44-.42.82-.72 1.13s-.64.54-1.04.7-.81.24-1.24.24-.84-.08-1.23-.24-.74-.39-1.04-.7c-.31-.31-.55-.69-.72-1.13-.18-.45-.27-.96-.27-1.53zm1.65 0c0 .69.14 1.24.43 1.66.28.41.68.62 1.18.62.51 0 .9-.21 1.19-.62.29-.42.44-.97.44-1.66 0-.7-.15-1.26-.44-1.67-.29-.42-.68-.63-1.19-.63-.5 0-.9.21-1.18.63-.29.41-.43.97-.43 1.67zm6.48-3.44h1.33l.12 1.21h.05c.24-.44.54-.79.88-1.02.35-.24.7-.36 1.07-.36.32 0 .59.05.78.14l-.28 1.4-.33-.09c-.11-.01-.23-.02-.38-.02-.27 0-.56.1-.86.31s-.55.58-.77 1.1v4.2h-1.61zm-47.87 15h1.61v4.1c0 .57.08.97.25 1.2.17.24.44.35.81.35.3 0 .57-.07.8-.22.22-.15.47-.39.73-.73v-4.7h1.61v6.87h-1.32l-.12-1.01h-.04c-.3.36-.63.64-.98.86-.35.21-.76.32-1.24.32-.73 0-1.27-.24-1.61-.71-.33-.47-.5-1.14-.5-2.02zm9.46 7.43v2.16h-1.61v-9.59h1.33l.12.72h.05c.29-.24.61-.45.97-.63.35-.17.72-.26 1.1-.26.43 0 .81.08 1.15.24.33.17.61.4.84.71.24.31.41.68.53 1.11.13.42.19.91.19 1.44 0 .59-.09 1.11-.25 1.57-.16.47-.38.85-.65 1.16-.27.32-.58.56-.94.73-.35.16-.72.25-1.1.25-.3 0-.6-.07-.9-.2s-.59-.31-.87-.56zm0-2.3c.26.22.5.37.73.45.24.09.46.13.66.13.46 0 .84-.2 1.15-.6.31-.39.46-.98.46-1.77 0-.69-.12-1.22-.35-1.61-.23-.38-.61-.57-1.13-.57-.49 0-.99.26-1.52.77zm5.87-1.69c0-.56.08-1.06.25-1.51.16-.45.37-.83.65-1.14.27-.3.58-.54.93-.71s.71-.25 1.08-.25c.39 0 .73.07 1 .2.27.14.54.32.81.55l-.06-1.1v-2.49h1.61v9.88h-1.33l-.11-.74h-.06c-.25.25-.54.46-.88.64-.33.18-.69.27-1.06.27-.87 0-1.56-.32-2.07-.95s-.76-1.51-.76-2.65zm1.67-.01c0 .74.13 1.31.4 1.7.26.38.65.58 1.15.58.51 0 .99-.26 1.44-.77v-3.21c-.24-.21-.48-.36-.7-.45-.23-.08-.46-.12-.7-.12-.45 0-.82.19-1.13.59-.31.39-.46.95-.46 1.68zm6.35 1.59c0-.73.32-1.3.97-1.71.64-.4 1.67-.68 3.08-.84 0-.17-.02-.34-.07-.51-.05-.16-.12-.3-.22-.43s-.22-.22-.38-.3c-.15-.06-.34-.1-.58-.1-.34 0-.68.07-1 .2s-.63.29-.93.47l-.59-1.08c.39-.24.81-.45 1.28-.63.47-.17.99-.26 1.54-.26.86 0 1.51.25 1.93.76s.63 1.25.63 2.21v4.07h-1.32l-.12-.76h-.05c-.3.27-.63.48-.98.66s-.73.27-1.14.27c-.61 0-1.1-.19-1.48-.56-.38-.36-.57-.85-.57-1.46zm1.57-.12c0 .3.09.53.27.67.19.14.42.21.71.21.28 0 .54-.07.77-.2s.48-.31.73-.56v-1.54c-.47.06-.86.13-1.18.23-.31.09-.57.19-.76.31s-.33.25-.41.4c-.09.15-.13.31-.13.48zm6.29-3.63h-.98v-1.2l1.06-.07.2-1.88h1.34v1.88h1.75v1.27h-1.75v3.28c0 .8.32 1.2.97 1.2.12 0 .24-.01.37-.04.12-.03.24-.07.34-.11l.28 1.19c-.19.06-.4.12-.64.17-.23.05-.49.08-.76.08-.4 0-.74-.06-1.02-.18-.27-.13-.49-.3-.67-.52-.17-.21-.3-.48-.37-.78-.08-.3-.12-.64-.12-1.01zm4.36 2.17c0-.56.09-1.06.27-1.51s.41-.83.71-1.14c.29-.3.63-.54 1.01-.71.39-.17.78-.25 1.18-.25.47 0 .88.08 1.23.24.36.16.65.38.89.67s.42.63.54 1.03c.12.41.18.84.18 1.32 0 .32-.02.57-.07.76h-4.37c.08.62.29 1.1.65 1.44.36.33.82.5 1.38.5.3 0 .58-.04.84-.13.25-.09.51-.21.76-.37l.54 1.01c-.32.21-.69.39-1.09.53s-.82.21-1.26.21c-.47 0-.92-.08-1.33-.25-.41-.16-.77-.4-1.08-.7-.3-.31-.54-.69-.72-1.13-.17-.44-.26-.95-.26-1.52zm4.61-.62c0-.55-.11-.98-.34-1.28-.23-.31-.58-.47-1.06-.47-.41 0-.77.15-1.08.45-.31.29-.5.73-.57 1.3zm3.01 2.23c.31.24.61.43.92.57.3.13.63.2.98.2.38 0 .65-.08.83-.23s.27-.35.27-.6c0-.14-.05-.26-.13-.37-.08-.1-.2-.2-.34-.28-.14-.09-.29-.16-.47-.23l-.53-.22c-.23-.09-.46-.18-.69-.3-.23-.11-.44-.24-.62-.4s-.33-.35-.45-.55c-.12-.21-.18-.46-.18-.75 0-.61.23-1.1.68-1.49.44-.38 1.06-.57 1.83-.57.48 0 .91.08 1.29.25s.71.36.99.57l-.74.98c-.24-.17-.49-.32-.73-.42-.25-.11-.51-.16-.78-.16-.35 0-.6.07-.76.21-.17.15-.25.33-.25.54 0 .14.04.26.12.36s.18.18.31.26c.14.07.29.14.46.21l.54.19c.23.09.47.18.7.29s.44.24.64.4c.19.16.34.35.46.58.11.23.17.5.17.82 0 .3-.06.58-.17.83-.12.26-.29.48-.51.68-.23.19-.51.34-.84.45-.34.11-.72.17-1.15.17-.48 0-.95-.09-1.41-.27-.46-.19-.86-.41-1.2-.68z" fill="#535353"/></g></svg>"></a></div><div class="c-bibliographic-information__column"><h3 class="c-article__sub-heading" id="citeas">Cite this article</h3><p class="c-bibliographic-information__citation">Szumiło, K. Frames in cofibration categories. <i>J. Homotopy Relat. Struct.</i> <b>12</b>, 577–616 (2017). https://doi.org/10.1007/s40062-016-0139-x</p><p class="c-bibliographic-information__download-citation u-hide-print"><a data-test="citation-link" data-track="click" data-track-action="download article citation" data-track-label="link" data-track-external="" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1007/s40062-016-0139-x?format=refman&flavour=citation">Download citation<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p><ul class="c-bibliographic-information__list" data-test="publication-history"><li class="c-bibliographic-information__list-item"><p>Received<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2016-05-08">08 May 2016</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Accepted<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2016-06-21">21 June 2016</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Published<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2016-07-13">13 July 2016</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Issue Date<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2017-09">September 2017</time></span></p></li><li class="c-bibliographic-information__list-item c-bibliographic-information__list-item--full-width"><p><abbr title="Digital Object Identifier">DOI</abbr><span class="u-hide">: </span><span class="c-bibliographic-information__value">https://doi.org/10.1007/s40062-016-0139-x</span></p></li></ul><div data-component="share-box"><div class="c-article-share-box u-display-none" hidden=""><h3 class="c-article__sub-heading">Share this article</h3><p class="c-article-share-box__description">Anyone you share the following link with will be able to read this content:</p><button class="js-get-share-url c-article-share-box__button" type="button" id="get-share-url" data-track="click" data-track-label="button" data-track-external="" data-track-action="get shareable link">Get shareable link</button><div class="js-no-share-url-container u-display-none" hidden=""><p class="js-c-article-share-box__no-sharelink-info c-article-share-box__no-sharelink-info">Sorry, a shareable link is not currently available for this article.</p></div><div class="js-share-url-container u-display-none" hidden=""><p class="js-share-url c-article-share-box__only-read-input" id="share-url" data-track="click" data-track-label="button" data-track-action="select share url"></p><button class="js-copy-share-url c-article-share-box__button--link-like" type="button" id="copy-share-url" data-track="click" data-track-label="button" data-track-action="copy share url" data-track-external="">Copy to clipboard</button></div><p class="js-c-article-share-box__additional-info c-article-share-box__additional-info"> Provided by the Springer Nature SharedIt content-sharing initiative </p></div></div><h3 class="c-article__sub-heading">Keywords</h3><ul class="c-article-subject-list"><li class="c-article-subject-list__subject"><span><a href="/search?query=Homotopy%20theory&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">Homotopy theory</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Homotopy%20colimit&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">Homotopy colimit</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Quasicategory&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">Quasicategory</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Cofibration%20category&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">Cofibration category</a></span></li></ul><div data-component="article-info-list"></div></div></div></div></div></section> </div> </main> <div class="c-article-sidebar u-text-sm u-hide-print l-with-sidebar__sidebar" id="sidebar" data-container-type="reading-companion" data-track-component="reading companion"> <aside> <div class="app-card-service" data-test="article-checklist-banner"> <div> <a class="app-card-service__link" data-track="click_presubmission_checklist" data-track-context="article page top of reading companion" data-track-category="pre-submission-checklist" data-track-action="clicked article page checklist banner test 2 old version" data-track-label="link" href="https://beta.springernature.com/pre-submission?journalId=40062" data-test="article-checklist-banner-link"> <span class="app-card-service__link-text">Use our pre-submission checklist</span> <svg class="app-card-service__link-icon" aria-hidden="true" focusable="false"><use xlink:href="#icon-eds-i-arrow-right-small"></use></svg> </a> <p class="app-card-service__description">Avoid common mistakes on your manuscript.</p> </div> <div class="app-card-service__icon-container"> <svg class="app-card-service__icon" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-clipboard-check-medium"></use> </svg> </div> </div> <div data-test="collections"> </div> <div data-test="editorial-summary"> </div> <div class="c-reading-companion"> <div class="c-reading-companion__sticky" data-component="reading-companion-sticky" data-test="reading-companion-sticky"> <div class="c-reading-companion__panel c-reading-companion__sections c-reading-companion__panel--active" id="tabpanel-sections"> <div class="u-lazy-ad-wrapper u-mt-16 u-hide" data-component-mpu><div class="c-ad c-ad--300x250"> <div class="c-ad__inner"> <p class="c-ad__label">Advertisement</p> <div id="div-gpt-ad-MPU1" class="div-gpt-ad grade-c-hide" data-pa11y-ignore data-gpt data-gpt-unitpath="/270604982/springerlink/40062/article" data-gpt-sizes="300x250" data-test="MPU1-ad" data-gpt-targeting="pos=MPU1;articleid=s40062-016-0139-x;"> </div> </div> </div> </div> </div> <div class="c-reading-companion__panel c-reading-companion__figures c-reading-companion__panel--full-width" id="tabpanel-figures"></div> <div class="c-reading-companion__panel c-reading-companion__references c-reading-companion__panel--full-width" id="tabpanel-references"></div> </div> </div> </aside> </div> </div> </article> <div class="app-elements"> <div class="eds-c-header__expander eds-c-header__expander--search" id="eds-c-header-popup-search"> <h2 class="eds-c-header__heading">Search</h2> <div class="u-container"> <search class="eds-c-header__search" role="search" aria-label="Search from the header"> <form method="GET" action="//link.springer.com/search" data-test="header-search" data-track="search" data-track-context="search from header" data-track-action="submit search form" data-track-category="unified header" data-track-label="form" > <label for="eds-c-header-search" class="eds-c-header__search-label">Search by keyword or author</label> <div class="eds-c-header__search-container"> <input id="eds-c-header-search" class="eds-c-header__search-input" autocomplete="off" name="query" type="search" value="" required> <button class="eds-c-header__search-button" type="submit"> <svg class="eds-c-header__icon" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-search-medium"></use> </svg> <span class="u-visually-hidden">Search</span> </button> </div> </form> </search> </div> </div> <div class="eds-c-header__expander eds-c-header__expander--menu" id="eds-c-header-nav"> <h2 class="eds-c-header__heading">Navigation</h2> <ul class="eds-c-header__list"> <li class="eds-c-header__list-item"> <a class="eds-c-header__link" href="https://link.springer.com/journals/" data-track="nav_find_a_journal" data-track-context="unified header" data-track-action="click find a journal" data-track-category="unified header" data-track-label="link" > Find a journal </a> </li> <li class="eds-c-header__list-item"> <a class="eds-c-header__link" href="https://www.springernature.com/gp/authors" data-track="nav_how_to_publish" data-track-context="unified header" data-track-action="click publish with us link" data-track-category="unified header" data-track-label="link" > Publish with us </a> </li> <li class="eds-c-header__list-item"> <a class="eds-c-header__link" href="https://link.springernature.com/home/" data-track="nav_track_your_research" data-track-context="unified header" data-track-action="click track your research" data-track-category="unified header" data-track-label="link" > Track your research </a> </li> </ul> </div> <footer > <div class="eds-c-footer" > <div class="eds-c-footer__container"> <div class="eds-c-footer__grid eds-c-footer__group--separator"> <div class="eds-c-footer__group"> <h3 class="eds-c-footer__heading">Discover content</h3> <ul class="eds-c-footer__list"> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://link.springer.com/journals/a/1" data-track="nav_journals_a_z" data-track-action="journals a-z" data-track-context="unified footer" data-track-label="link">Journals A-Z</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://link.springer.com/books/a/1" data-track="nav_books_a_z" data-track-action="books a-z" data-track-context="unified footer" data-track-label="link">Books A-Z</a></li> </ul> </div> <div class="eds-c-footer__group"> <h3 class="eds-c-footer__heading">Publish with us</h3> <ul class="eds-c-footer__list"> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://link.springer.com/journals" data-track="nav_journal_finder" data-track-action="journal finder" data-track-context="unified footer" data-track-label="link">Journal finder</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/authors" data-track="nav_publish_your_research" data-track-action="publish your research" data-track-context="unified footer" data-track-label="link">Publish your research</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/open-research/about/the-fundamentals-of-open-access-and-open-research" data-track="nav_open_access_publishing" data-track-action="open access publishing" data-track-context="unified footer" data-track-label="link">Open access publishing</a></li> </ul> </div> <div class="eds-c-footer__group"> <h3 class="eds-c-footer__heading">Products and services</h3> <ul class="eds-c-footer__list"> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/products" data-track="nav_our_products" data-track-action="our products" data-track-context="unified footer" data-track-label="link">Our products</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/librarians" data-track="nav_librarians" data-track-action="librarians" data-track-context="unified footer" data-track-label="link">Librarians</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/societies" data-track="nav_societies" data-track-action="societies" data-track-context="unified footer" data-track-label="link">Societies</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/partners" data-track="nav_partners_and_advertisers" data-track-action="partners and advertisers" data-track-context="unified footer" data-track-label="link">Partners and advertisers</a></li> </ul> </div> <div class="eds-c-footer__group"> <h3 class="eds-c-footer__heading">Our imprints</h3> <ul class="eds-c-footer__list"> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springer.com/" data-track="nav_imprint_Springer" data-track-action="Springer" data-track-context="unified footer" data-track-label="link">Springer</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.nature.com/" data-track="nav_imprint_Nature_Portfolio" data-track-action="Nature Portfolio" data-track-context="unified footer" data-track-label="link">Nature Portfolio</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.biomedcentral.com/" data-track="nav_imprint_BMC" data-track-action="BMC" data-track-context="unified footer" data-track-label="link">BMC</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.palgrave.com/" data-track="nav_imprint_Palgrave_Macmillan" data-track-action="Palgrave Macmillan" data-track-context="unified footer" data-track-label="link">Palgrave Macmillan</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.apress.com/" data-track="nav_imprint_Apress" data-track-action="Apress" data-track-context="unified footer" data-track-label="link">Apress</a></li> </ul> </div> </div> </div> <div class="eds-c-footer__container"> <nav aria-label="footer navigation"> <ul class="eds-c-footer__links"> <li class="eds-c-footer__item"> <button class="eds-c-footer__link" data-cc-action="preferences" data-track="dialog_manage_cookies" data-track-action="Manage cookies" data-track-context="unified footer" data-track-label="link"><span class="eds-c-footer__button-text">Your privacy choices/Manage cookies</span></button> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://www.springernature.com/gp/legal/ccpa" data-track="nav_california_privacy_statement" data-track-action="california privacy statement" data-track-context="unified footer" data-track-label="link">Your US state privacy rights</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://www.springernature.com/gp/info/accessibility" data-track="nav_accessibility_statement" data-track-action="accessibility statement" data-track-context="unified footer" data-track-label="link">Accessibility statement</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://link.springer.com/termsandconditions" data-track="nav_terms_and_conditions" data-track-action="terms and conditions" data-track-context="unified footer" data-track-label="link">Terms and conditions</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://link.springer.com/privacystatement" data-track="nav_privacy_policy" data-track-action="privacy policy" data-track-context="unified footer" data-track-label="link">Privacy policy</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://support.springernature.com/en/support/home" data-track="nav_help_and_support" data-track-action="help and support" data-track-context="unified footer" data-track-label="link">Help and support</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://link.springer.com/legal-notice" data-track="nav_legal_notice" data-track-action="legal notice" data-track-context="unified footer" data-track-label="link">Legal notice</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://support.springernature.com/en/support/solutions/articles/6000255911-subscription-cancellations" data-track-action="cancel contracts here">Cancel contracts here</a> </li> </ul> </nav> <div class="eds-c-footer__user"> <p class="eds-c-footer__user-info"> <span data-test="footer-user-ip">8.222.208.146</span> </p> <p class="eds-c-footer__user-info" data-test="footer-business-partners">Not affiliated</p> </div> <a href="https://www.springernature.com/" class="eds-c-footer__link"> <img src="/oscar-static/images/logo-springernature-white-19dd4ba190.svg" alt="Springer Nature" loading="lazy" width="200" height="20"/> </a> <p class="eds-c-footer__legal" data-test="copyright">© 2025 Springer Nature</p> </div> </div> </footer> </div> </body> </html>