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12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> Brown representability theorem </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/4053/#Item_14" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" 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class="existingWikiWord" href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%3Fech+cohomology">?ech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="topos_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-Topos Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a></strong></p> <h2 id="sidebar_background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf">(∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elementary+%28%E2%88%9E%2C1%29-topos">elementary (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">localization of an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+localization">topological localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercompletion">hypercompletion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>/<a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-topos">(n,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/n-topos">n-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/truncated">n-truncated object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected">n-connected object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">(1,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-presheaf">(2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-quasitopos">(∞,1)-quasitopos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+%28%E2%88%9E%2C1%29-presheaf">separated (∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+presheaf">separated presheaf</a></li> </ul> </li> <li> <p><span class="newWikiWord">(2,1)-quasitopos<a href="/nlab/new/%282%2C1%29-quasitopos">?</a></span></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+%282%2C1%29-presheaf">separated (2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-topos">(∞,2)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-topos">(∞,n)-topos</a></p> </li> </ul> <h2 id="sidebar_characterization">Characterization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object in an (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/effective+epimorphism">effective epimorphism</a></li> </ul> </li> </ul> <h2 id="sidebar_morphisms">Morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-geometric+morphism">(∞,1)-geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Topos">(∞,1)Topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lawvere+distribution">Lawvere distribution</a></p> </li> </ul> <h2 id="sidebar_extra">Extra stuff, structure and property</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+%28%E2%88%9E%2C1%29-topos">hypercomplete (∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+object">hypercomplete object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+theorem">Whitehead theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over-%28%E2%88%9E%2C1%29-topos">over-(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-localic+%28%E2%88%9E%2C1%29-topos">n-localic (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+n-connected+%28n%2C1%29-topos">locally n-connected (n,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/concrete+%28%E2%88%9E%2C1%29-sheaf">concrete (∞,1)-sheaf</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></p> </li> </ul> <h2 id="sidebar_models">Models</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+functors">model structure on functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+site">model site</a>/<a class="existingWikiWord" href="/nlab/show/sSet-site">sSet-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent+for+simplicial+presheaves">descent for simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Verity+on+descent+for+strict+omega-groupoid+valued+presheaves">descent for presheaves with values in strict ∞-groupoids</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_constructions">Constructions</h2> <p><strong>structures in a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/shape+of+an+%28%E2%88%9E%2C1%29-topos">shape</a> / <a class="existingWikiWord" href="/nlab/show/coshape+of+an+%28%E2%88%9E%2C1%29-topos">coshape</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a>/<a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical</a>/<a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric</a> homotopy groups</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Postnikov+tower+in+an+%28%E2%88%9E%2C1%29-category">Postnikov tower</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+tower+in+an+%28%E2%88%9E%2C1%29-topos">Whitehead tower</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory+in+an+%28%E2%88%9E%2C1%29-topos">rational homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+dimension">homotopy dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomological+dimension">cohomological dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covering+dimension">covering dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+dimension">Heyting dimension</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/%28infinity%2C1%29-topos+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#ClassicalFormulation'>Classical formulation for homotopy functors on topological spaces</a></li> <li><a href='#InInfinity1Categories'>For homotopy functors on presentable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</a></li> <li><a href='#further_variants'>Further variants</a></li> <ul> <li><a href='#for_triangulated_categories_and_model_categories'>For triangulated categories and model categories</a></li> <li><a href='#ForEquivariantCohomology'>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RO</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">RO(G)</annotation></semantics></math>-graded equivariant cohomology</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#multiplicative_cohomology_and_ring_spectra'>Multiplicative cohomology and ring spectra</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The classical <em>Brown representability theorem</em> (<a href="#Brown62">Brown 62</a>, <a href="#Adams71">Adams 71</a>) says contravariant <a class="existingWikiWord" href="/nlab/show/functors">functors</a> on the (pointed) <a class="existingWikiWord" href="/nlab/show/classical+homotopy+category">classical homotopy category</a> satisfying two conditions (“<a class="existingWikiWord" href="/nlab/show/Brown+functors">Brown functors</a>”) are <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a>.</p> <p>This is used notably to show that every additive reduced <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">E^\bullet</annotation></semantics></math> is representable by a <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><msub><mi>E</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E^n(X) \simeq [X,E_n] \,. </annotation></semantics></math></div> <p>(But beware that the cohomology theory in general contains less information than the spectrum, due to <a class="existingWikiWord" href="/nlab/show/phantom+maps">phantom maps</a> (see also this <a href="http://mathoverflow.net/q/117684/381">MO discussion</a>).)</p> <p>The Brown representability theorem as such, with the two conditions on a <a class="existingWikiWord" href="/nlab/show/Brown+functor">Brown functor</a> understood, only applies to <a class="existingWikiWord" href="/nlab/show/contravariant+functors">contravariant functors</a>, not to covariant functors. But by way of <a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+duality">Spanier-Whitehead duality</a> it implies at least over <a class="existingWikiWord" href="/nlab/show/finite+CW-complexes">finite CW-complexes</a> that dually every additive <a class="existingWikiWord" href="/nlab/show/generalized+homology+theory">generalized homology theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">E_\bullet</annotation></semantics></math> is representable by a spectrum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> via</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>n</mi></msub><mo>∧</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E_n(X) \simeq \pi_n(E_n \wedge X) \,. </annotation></semantics></math></div> <p>There are various generalizations, such as to <a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated categories</a> (<a href="#Neeman96">Neeman 96</a>), to <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy categories of</a> various <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> (<a href="#Jardine09">Jardine 09</a>) and <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category">homotopy categories of</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a> (<a href="#LurieHigherAlgebra">Lurie, theorem 1.4.1.2</a>). But in any case there are some crucial conditions for the theorem to apply, such as</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a> structure on the values of the functor, as is the case for an (abelian) cohomology theory, and as would be the case for a represented functor in any <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a>,</li> </ul> <p>OR</p> <ul> <li>the existence of a <a class="existingWikiWord" href="/nlab/show/strong+generator">strong generator</a> in the homotopy category in question.</li> </ul> <p>In particular, there is no Brown representability theorem for functors from the homotopy category of pointed not-necessarily-connected spaces to pointed sets, or for functors from the homotopy category of unpointed spaces to sets. In fact, there are counterexamples in these two cases (<a href="#FreydHeller93">Freyd-Heller 93</a>, see remark <a class="maruku-ref" href="#Counterexamples"></a>).</p> <h2 id="ClassicalFormulation">Classical formulation for homotopy functors on topological spaces</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mo>*</mo> <mi>c</mi></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(Top_*^c)</annotation></semantics></math> denote the <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> of connected <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a> under <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> — or equivalently the homotopy category of pointed connected <a class="existingWikiWord" href="/nlab/show/CW+complexes">CW complexes</a> under <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mo>*</mo> <mi>c</mi></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(Top_*^c)</annotation></semantics></math> has <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> (given by <a class="existingWikiWord" href="/nlab/show/wedge+sums">wedge sums</a>) and also <a class="existingWikiWord" href="/nlab/show/weak+limit">weak pushouts</a> (namely, <a class="existingWikiWord" href="/nlab/show/homotopy+pushouts">homotopy pushouts</a>, see at <a href="weak+limit#RelationToHomotopyLimits">weak limit</a>).</p> <div class="num_theorem" id="ClassicalBrownRepresentabilityTheorem"> <h6 id="theorem">Theorem</h6> <p><strong>(Brown-Adams)</strong></p> <p>A <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>Ho</mi><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mo>*</mo> <mi>c</mi></msubsup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><msub><mi>Set</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">F:Ho(Top_*^c)^{op} \to Set_*</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a> precisely if</p> <ol> <li> <p>it <a class="existingWikiWord" href="/nlab/show/preserved+limit">takes</a> <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> to <a class="existingWikiWord" href="/nlab/show/products">products</a>,</p> </li> <li> <p>it takes <a class="existingWikiWord" href="/nlab/show/weak+pushouts">weak pushouts</a> to <a class="existingWikiWord" href="/nlab/show/weak+pullbacks">weak pullbacks</a>.</p> </li> </ol> </div> <p>(e.g. <a href="#AguilarGitlerPrito02">Aguilar-Gitler-Prito 02, theorem 12.2.18</a>).</p> <p>Note that it is immediate that every representable functor has the given properties; the nontrivial statement is that these properties already characterize representable functors.</p> <p>When the theorem is stated in terms of CW complexes, the second property (taking <a class="existingWikiWord" href="/nlab/show/weak+pushouts">weak pushouts</a> to <a class="existingWikiWord" href="/nlab/show/weak+pullbacks">weak pullbacks</a>) is often phrased equivalently as:</p> <ul> <li>The <strong>Mayer-Vietoris axiom</strong>: For every <a class="existingWikiWord" href="/nlab/show/CW-pair">CW-triple</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X; A_1, A_2)</annotation></semantics></math> (with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><mo>∪</mo><msub><mi>A</mi> <mn>2</mn></msub><mo>=</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A_1\cup A_2 = X</annotation></semantics></math>) and any elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>∈</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x_1\in F(A_1)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>2</mn></msub><mo>∈</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x_2\in F(A_2)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">|</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>A</mi> <mn>2</mn></msub><mo>=</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">|</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_1|A_1\cap A_2 = x_2|A_1\cap A_2</annotation></semantics></math>, there exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y\in F(X)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo stretchy="false">|</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">y|A_1 = x_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo stretchy="false">|</mo><msub><mi>A</mi> <mn>2</mn></msub><mo>=</mo><msub><mi>x</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">y|A_2 = x_2</annotation></semantics></math>.</li> </ul> <div class="num_remark" id="Counterexamples"> <h6 id="remark">Remark</h6> <p><strong>(counterexamples)</strong></p> <p>The statement of theorem <a class="maruku-ref" href="#ClassicalBrownRepresentabilityTheorem"></a> without the restriction to <em>connected</em> pointed spaces is false, as is the analogous statement for unpointed spaces.</p> <p>In (<a href="#FreydHeller93">Freyd-Heller 93</a>), it is shown that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/Thompson%27s+group+F">Thompson's group F</a>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g:G\to G</annotation></semantics></math> its canonical endomorphism, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> does not <a class="existingWikiWord" href="/nlab/show/split+idempotent">split</a> in the quotient of <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a> by conjugacy. Since the quotient of <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a> by conjugacy embeds as the full subcategory of the unpointed homotopy category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>Top</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(Top)</annotation></semantics></math> on connected <a class="existingWikiWord" href="/nlab/show/homotopy+1-types">homotopy 1-types</a>, we have an endomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>g</mi><mo>:</mo><mi>B</mi><mi>G</mi><mo>→</mo><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">B g:B G \to B G</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> which does not split in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>Top</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(Top)</annotation></semantics></math>.</p> <p>Thus, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>Top</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">F:Ho(Top)^{op} \to Set</annotation></semantics></math> splits the idempotent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>B</mi><mi>g</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,B g]</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>B</mi><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,B G]</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> satisfies the hypotheses of the Brown representability theorem (being a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of a representable functor), but is not representable. A similar argument using <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><msub><mi>G</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">B G_+</annotation></semantics></math> applies to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msub><mi>Top</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(Top_*)</annotation></semantics></math>.</p> <p>There is also another example due to Heller, which fails to be representable for cardinality reasons.</p> <ul> <li>MO questions <a href="http://mathoverflow.net/questions/104866/brown-representability-for-non-connected-spaces">non-connected spaces</a>, <a href="http://mathoverflow.net/questions/11456/unpointed-brown-representability-theorem">unpointed spaces</a></li> <li><a href="http://golem.ph.utexas.edu/category/2012/08/brown_representability.html">blog post</a></li> </ul> </div> <h2 id="InInfinity1Categories">For homotopy functors on presentable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</h2> <p>The <a href="#ClassicalFormulation">classical formulation</a> of Brown representability is only superficially concerned with <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>. Instead, via the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a>, one finds that the statement only concerns the <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a> of <a class="existingWikiWord" href="/nlab/show/Top">Top</a> at the <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a>, hence the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>whe</mi></msub><mi>Top</mi><mo>≃</mo></mrow><annotation encoding="application/x-tex">L_{whe} Top\simeq </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>.</p> <p>We discuss now the natural formulation of the Brown representability theorem for functors out of <a class="existingWikiWord" href="/nlab/show/homotopy+categories+of+%28%E2%88%9E%2C1%29-categories">homotopy categories of (∞,1)-categories</a> following (<a href="#LurieHigherAlgebra">Lurie, section 1.4.1</a>). See also the exposition in (<a href="#Mathew11">Mathew 11</a>).</p> <div class="num_defn" id="BrownFunctorOnInfinityCategory"> <h6 id="definition">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable (∞,1)-category</a>. A <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>⟶</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> F \;\colon\; Ho(\mathcal{C})^{op} \longrightarrow Set </annotation></semantics></math></div> <p>(from the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite</a> of the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category">homotopy category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> to <a class="existingWikiWord" href="/nlab/show/Set">Set</a>)</p> <p>is called a <strong><a class="existingWikiWord" href="/nlab/show/Brown+functor">Brown functor</a></strong> if</p> <ol> <li> <p>it sends small <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> to <a class="existingWikiWord" href="/nlab/show/products">products</a>;</p> </li> <li> <p>it sends <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pushouts">(∞,1)-pushouts</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>→</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}\to Ho(\mathcal{C})</annotation></semantics></math> to <a class="existingWikiWord" href="/nlab/show/weak+pullbacks">weak pullbacks</a> in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>.</p> </li> </ol> </div> <div class="num_remark" id="WeakPullbacks"> <h6 id="remark_2">Remark</h6> <p>A <em><a class="existingWikiWord" href="/nlab/show/weak+pullback">weak pullback</a></em> is a diagram that satisfies the existence clause of a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>, but not necessarily the uniqueness condition. Hence the second clause in def. <a class="maruku-ref" href="#BrownFunctorOnInfinityCategory"></a> says that for a <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pushout">(∞,1)-pushout</a> square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Z</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><munder><mo>⊔</mo><mi>Z</mi></munder><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Z &\longrightarrow& X \\ \downarrow &\swArrow& \downarrow \\ Y &\longrightarrow& X \underset{Z}{\sqcup}Y } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, then the induced universal morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mrow><mo>(</mo><mi>X</mi><munder><mo>⊔</mo><mi>Z</mi></munder><mi>Y</mi><mo>)</mo></mrow><mover><mo>⟶</mo><mi>epi</mi></mover><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><munder><mo>×</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> F\left(X \underset{Z}{\sqcup}Y\right) \stackrel{epi}{\longrightarrow} F(X) \underset{F(Z)}{\times} F(Y) </annotation></semantics></math></div> <p>into the actual <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>.</p> </div> <div class="num_defn" id="CompactGenerationByCogroupObjects"> <h6 id="definition_2">Definition</h6> <p>Say that a <a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable (∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is <strong>compactly generated by cogroup objects closed under suspensions</strong> if</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/compactly+generated+%28%E2%88%9E%2C1%29-category">generated</a> by a set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>∈</mo><mi>𝒞</mi><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \{S_i \in \mathcal{C}\}_{i \in I} </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/compact+object+in+an+%28infinity%2C1%29-category">compact objects</a> (i.e. every object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimit">(∞,1)-colimit</a> of the objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">S_i</annotation></semantics></math>.)</p> </li> <li> <p>each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">S_i</annotation></semantics></math> admits the structure of a <a class="existingWikiWord" href="/nlab/show/cogroup">cogroup</a> object in the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category">homotopy category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C})</annotation></semantics></math>;</p> </li> <li> <p>the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>S</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{S_i\}</annotation></semantics></math> is closed under forming <a class="existingWikiWord" href="/nlab/show/reduced+suspensions">reduced suspensions</a>.</p> </li> </ol> </div> <div class="num_example" id="SuspensionsAreHCogroupObjects"> <h6 id="example">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/suspensions+are+H-cogroup+objects">suspensions are H-cogroup objects</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> with <a class="existingWikiWord" href="/nlab/show/finite+%28%E2%88%9E%2C1%29-colimits">finite (∞,1)-colimits</a> and with a <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>↦</mo><mn>0</mn><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>X</mi></munder><mn>0</mn></mrow><annotation encoding="application/x-tex">\Sigma \colon X \mapsto 0 \underset{X}{\coprod} 0</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/reduced+suspension">reduced suspension</a> functor.</p> <p>Then the <a class="existingWikiWord" href="/nlab/show/fold+map">fold map</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mi>X</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>Σ</mi><mi>X</mi><mo>≃</mo><mn>0</mn><munder><mo>⊔</mo><mi>X</mi></munder><mn>0</mn><munder><mo>⊔</mo><mi>X</mi></munder><mn>0</mn><mo>⟶</mo><mn>0</mn><munder><mo>⊔</mo><mi>X</mi></munder><mi>X</mi><munder><mo>⊔</mo><mi>X</mi></munder><mn>0</mn><mo>≃</mo><mn>0</mn><munder><mo>⊔</mo><mi>X</mi></munder><mn>0</mn><mo>≃</mo><mi>Σ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex"> \Sigma X \coprod \Sigma X \simeq 0 \underset{X}{\sqcup} 0 \underset{X}{\sqcup} 0 \longrightarrow 0 \underset{X}{\sqcup} X \underset{X}{\sqcup} 0 \simeq 0 \underset{X}{\sqcup} 0 \simeq \Sigma X </annotation></semantics></math></div> <p>exhibits <a class="existingWikiWord" href="/nlab/show/cogroup">cogroup</a> structure on the image of any <a class="existingWikiWord" href="/nlab/show/suspension+object">suspension object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma X</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category</a>.</p> <p>This is equivalently the <a class="existingWikiWord" href="/nlab/show/group">group</a>-structure of the first (<a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental</a>) <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a> of the values of <a class="existingWikiWord" href="/nlab/show/representable+functor">functor co-represented</a> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma X</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Y</mi><mo>↦</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Ω</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>π</mi> <mn>1</mn></msub><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ho(\mathcal{C})(\Sigma X, -) \;\colon\; Y \mapsto Ho(\mathcal{C})(\Sigma X, Y) \simeq Ho(\mathcal{C})(X, \Omega Y) \simeq \pi_1 Ho(\mathcal{C})(X, Y) \,. </annotation></semantics></math></div></div> <div class="num_example" id="TheClassicalPointedConnectedHomotopyCategoryAsDomainForTheAbstractBrownRepresentabilityTheorem"> <h6 id="example_2">Example</h6> <p>In bare <a class="existingWikiWord" href="/nlab/show/pointed+homotopy+types">pointed homotopy types</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{C} = </annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mrow></mrow> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">{}^{\ast/}</annotation></semantics></math>, the (<a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> of) <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^n</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/cogroup">cogroup</a> objects for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq 1</annotation></semantics></math>, but not for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math>, by example <a class="maruku-ref" href="#SuspensionsAreHCogroupObjects"></a>. And of course they are <a class="existingWikiWord" href="/nlab/show/compact+object+in+an+%28%E2%88%9E%2C1%29-category">compact objects</a>.</p> <p>So while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>S</mi> <mi>n</mi></msup><msub><mo stretchy="false">}</mo> <mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{S^n\}_{n \in \mathbb{N}}</annotation></semantics></math> generates all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><msup><mi>Grpd</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\infty Grpd^{\ast/}</annotation></semantics></math>, the latter is <em>not</em> an example of def. <a class="maruku-ref" href="#CompactGenerationByCogroupObjects"></a> due to the failure of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">S^0</annotation></semantics></math> to have <a class="existingWikiWord" href="/nlab/show/cogroup">cogroup</a> structure.</p> <p>Removing that generator, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category generated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>S</mi> <mi>n</mi></msup><msub><mo stretchy="false">}</mo> <mfrac linethickness="0"><mrow><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></mrow><mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></mrow></mfrac></msub></mrow><annotation encoding="application/x-tex">\{S^n\}_{{n \in \mathbb{N}} \atop {n \geq 1}}</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><msubsup><mi>Grpd</mi> <mrow><mo>≥</mo><mn>1</mn></mrow> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\infty Grpd^{\ast/}_{\geq 1}</annotation></semantics></math>, that of <em><a class="existingWikiWord" href="/nlab/show/connected+object">connected</a></em> <a class="existingWikiWord" href="/nlab/show/pointed+homotopy+types">pointed homotopy types</a>. This is one way to see how the connectedness condition in the classical version of Brown representability theorem arises.</p> </div> <p>See also (<a href="#LurieHigherAlgebra">Lurie, example 1.4.1.4</a>)</p> <p>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categories compactly generated by cogroup objects closed under forming suspensions, the following strenghtening of the <a class="existingWikiWord" href="/nlab/show/Whitehead+theorem">Whitehead theorem</a> holds.</p> <div class="num_prop" id="WhiteheadTheoremForCompactGenerationByCogroupObjects"> <h6 id="proposition">Proposition</h6> <p>In an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category compactly generated by cogroup objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>S</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{S_i\}_{i \in I}</annotation></semantics></math> closed under forming suspensions, according to def. <a class="maruku-ref" href="#CompactGenerationByCogroupObjects"></a>, a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X \longrightarrow Y</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28infinity%2C1%29-category">equivalence</a> precisely if for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i \in I</annotation></semantics></math> the induced function of maps in the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category">homotopy category</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ho(\mathcal{C})(S_i,f) \;\colon\; Ho(\mathcal{C})(S_i,X) \longrightarrow Ho(\mathcal{C})(S_i,Y) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> (a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a>).</p> </div> <p>(<a href="#LurieHigherAlgebra">Lurie, p. 114, Lemma star</a>)</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>By the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a>, the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28%E2%88%9E%2C1%29-category">equivalence</a> precisely if for all objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">A \in \mathcal{C}</annotation></semantics></math> the induced morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{C}(A,f) \;\colon\; \mathcal{C}(A,X) \longrightarrow \mathcal{C}(A,Y) </annotation></semantics></math></div> <p>is an equivalence in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>. By assumption of compact generation and since the hom-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(-,-)</annotation></semantics></math> sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-colimits in the first argument to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-limits, this is the case precisely already if it is the case for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mo stretchy="false">{</mo><msub><mi>S</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">A \in \{S_i\}_{i \in I}</annotation></semantics></math>.</p> <p>Now by the standard <a class="existingWikiWord" href="/nlab/show/Whitehead+theorem">Whitehead theorem</a> in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> (being a <a class="existingWikiWord" href="/nlab/show/hypercomplete+%28%E2%88%9E%2C1%29-topos">hypercomplete (∞,1)-topos</a>), the morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>𝒞</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{C}(S_i,f) \;\colon\; \mathcal{C}(S_i,X) \longrightarrow \mathcal{C}(S_i,Y) </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> are <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28%E2%88%9E%2C1%29-category">equivalences</a> precisely if they are <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a>, hence precisely if they induce <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> on all <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\pi_n</annotation></semantics></math> for <strong>all basepoints</strong>.</p> <p>It is this last condition of testing on all basepoints that the assumed <a class="existingWikiWord" href="/nlab/show/cogroup">cogroup</a> structure on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">S_i</annotation></semantics></math> allows to do away with: this cogroup structure implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(S_i,-)</annotation></semantics></math> has the structure of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>-group, and this implies (by group multiplication), that all <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a> have the same homotopy groups, hence that all homotopy groups are independent of the choice of basepoint, up to isomorphism.</p> <p>Therefore the above morphisms are equivalences precisely if they are so under applying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\pi_n</annotation></semantics></math> based on the connected component of the <a class="existingWikiWord" href="/nlab/show/zero+morphism">zero morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mi>𝒞</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>π</mi> <mi>n</mi></msub><mi>𝒞</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mi>n</mi></msub><mi>𝒞</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_n\mathcal{C}(S_i,f) \;\colon\; \pi_n \mathcal{C}(S_i,X) \longrightarrow \pi_n\mathcal{C}(S_i,Y) \,. </annotation></semantics></math></div> <p>Now in this pointed situation we may use that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>π</mi> <mi>n</mi></msub><mi>𝒞</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><msub><mi>π</mi> <mn>0</mn></msub><mi>𝒞</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>π</mi> <mn>0</mn></msub><mi>𝒞</mi><mo stretchy="false">(</mo><msup><mi>Σ</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>Σ</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \pi_n \mathcal{C}(-,-) & \simeq \pi_0 \mathcal{C}(-,\Omega^n(-)) \\ & \simeq \pi_0\mathcal{C}(\Sigma^n(-),-) \\ & \simeq Ho(\mathcal{C})(\Sigma^n(-),-) \end{aligned} </annotation></semantics></math></div> <p>to find that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is an equivalence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> precisely if the induced morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>Σ</mi> <mi>n</mi></msup><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>Σ</mi> <mi>n</mi></msup><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>Σ</mi> <mi>n</mi></msup><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ho(\mathcal{C})(\Sigma^n S_i, f) \;\colon\; Ho(\mathcal{C})(\Sigma^n S_i,X) \longrightarrow Ho(\mathcal{C})(\Sigma^n S_i,Y) </annotation></semantics></math></div> <p>are isomorphisms for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i \in I</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>.</p> <p>Finally by the assumption that each suspension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mi>n</mi></msup><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\Sigma^n S_i</annotation></semantics></math> of a generator is itself among the set of generators, the claim follows.</p> </div> <div class="num_theorem" id="BrownRepresentabilityOnPresentableInfinityCategories"> <h6 id="theorem_2">Theorem</h6> <p><strong>(Brown representability)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category compactly generated by cogroup objects closed under forming suspensions, according to def. <a class="maruku-ref" href="#CompactGenerationByCogroupObjects"></a>. Then a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>⟶</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> F \;\colon\; Ho(\mathcal{C})^{op} \longrightarrow Set </annotation></semantics></math></div> <p>(from the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite</a> of the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category">homotopy category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> to <a class="existingWikiWord" href="/nlab/show/Set">Set</a>) is <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a> precisely if it is a <a class="existingWikiWord" href="/nlab/show/Brown+functor">Brown functor</a>, def. <a class="maruku-ref" href="#BrownFunctorOnInfinityCategory"></a>.</p> </div> <p>(<a href="#LurieHigherAlgebra">Lurie, theorem 1.4.1.2</a>)</p> <div class="proof" id="ProofFollowingLurie"> <h6 id="proof_2">Proof</h6> <p>Due to the version of the Whitehead theorem of prop. <a class="maruku-ref" href="#WhiteheadTheoremForCompactGenerationByCogroupObjects"></a> we are essentially reduced to showing that <a class="existingWikiWord" href="/nlab/show/Brown+functors">Brown functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> are representable on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">S_i</annotation></semantics></math>. To that end consider the following lemma. (In the following we notationally identify, via the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>, objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, hence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C})</annotation></semantics></math>, with the functors they <a class="existingWikiWord" href="/nlab/show/representable+functor">represent</a>.)</p> <p>Lemma (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋆</mo></mrow><annotation encoding="application/x-tex">\star</annotation></semantics></math>): <em>Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>∈</mo><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta \in F(X)</annotation></semantics></math>, hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\eta \colon X \to F</annotation></semantics></math>, then there exists a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>X</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f \colon X \to X'</annotation></semantics></math> and an <a class="existingWikiWord" href="/nlab/show/extension">extension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>′</mo><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>′</mo><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\eta' \colon X' \to F</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> which induces for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">S_i</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>′</mo><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mi>X</mi><mo>′</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><mi>PSh</mi><mo stretchy="false">(</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta'\circ (-) \colon Ho(\mathcal{C})(S_i,X') \stackrel{\simeq}{\longrightarrow} PSh(Ho(\mathcal{C}))(S_i,F) \simeq F(S_i)</annotation></semantics></math>.</em></p> <p>To see this, first notice that we may directly find an extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\eta_0</annotation></semantics></math> along a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msub><mi>X</mi> <mi>o</mi></msub></mrow><annotation encoding="application/x-tex">X\to X_o</annotation></semantics></math> such as to make a <a class="existingWikiWord" href="/nlab/show/surjection">surjection</a>: simply take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math> to be the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> of <strong>all</strong> possible elements in the codomain and take</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mn>0</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⊔</mo><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mfrac linethickness="0"><mrow><mrow><mi>i</mi><mo>∈</mo><mi>I</mi><mo>,</mo></mrow></mrow><mrow><mrow><mi>γ</mi><mo lspace="verythinmathspace">:</mo><msub><mi>S</mi> <mi>i</mi></msub><mover><mo>→</mo><mrow></mrow></mover><mi>F</mi></mrow></mrow></mfrac></munder><msub><mi>S</mi> <mi>i</mi></msub><mo>)</mo></mrow><mo>⟶</mo><mi>F</mi></mrow><annotation encoding="application/x-tex"> \eta_0 \;\colon\; X \sqcup \left( \underset{{i \in I,} \atop {\gamma \colon S_i \stackrel{}{\to} F}}{\coprod} S_i \right) \longrightarrow F </annotation></semantics></math></div> <p>to be the canonical map. (Using that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>, by assumption, turns coproducts into products, we may indeed treat the coproduct in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> on the left as the coproduct of the corresponding functors.)</p> <p>To turn the surjection thus constructed into a bijection, we now successively form quotients of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math>. To that end proceed by <a class="existingWikiWord" href="/nlab/show/induction">induction</a> and suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>n</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\eta_n \colon X_n \to F</annotation></semantics></math> has been constructed. Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i \in I</annotation></semantics></math> let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>i</mi></msub><mo>≔</mo><mi>ker</mi><mrow><mo>(</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>n</mi></msub><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover><mi>F</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> K_i \coloneqq ker \left( Ho(\mathcal{C})(S_i, X_n) \stackrel{\eta_n \circ (-)}{\longrightarrow} F(S_i) \right) </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\eta_n</annotation></semantics></math> evaluated on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">S_i</annotation></semantics></math>. These <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">K_i</annotation></semantics></math> are the pieces that need to go away in order to make a bijection. Hence define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{n+1}</annotation></semantics></math> to be their joint <a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>≔</mo><mi>coker</mi><mrow><mo>(</mo><mrow><mo>(</mo><munder><mo>⊔</mo><mfrac linethickness="0"><mrow><mrow><mi>i</mi><mo>∈</mo><mi>I</mi><mo>,</mo></mrow></mrow><mrow><mrow><mi>γ</mi><mo>∈</mo><msub><mi>K</mi> <mi>i</mi></msub></mrow></mrow></mfrac></munder><msub><mi>S</mi> <mi>i</mi></msub><mo>)</mo></mrow><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>γ</mi><msub><mo stretchy="false">)</mo> <mfrac linethickness="0"><mrow><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></mrow><mrow><mrow><mi>γ</mi><mo>∈</mo><msub><mi>K</mi> <mi>i</mi></msub></mrow></mrow></mfrac></msub></mrow></mover><msub><mi>X</mi> <mi>n</mi></msub><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X_{n+1} \coloneqq coker\left( \left( \underset{{i \in I,} \atop {\gamma \in K_i}}{\sqcup} S_i \right) \overset{(\gamma)_{{i \in I} \atop {\gamma\in K_i}}}{\longrightarrow} X_n \right) \,. </annotation></semantics></math></div> <p>Then by the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> takes this homotopy cokernel to a <a class="existingWikiWord" href="/nlab/show/weak+limit">weak</a> <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> (as in remark <a class="maruku-ref" href="#WeakPullbacks"></a>), there exists an extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\eta_{n+1}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\eta_n</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_n \to X_{n+1}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mrow><mo>(</mo><munder><mo>⊔</mo><mfrac linethickness="0"><mrow><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></mrow><mrow><mrow><mi>γ</mi><mo>∈</mo><msub><mi>K</mi> <mi>i</mi></msub></mrow></mrow></mfrac></munder><msub><mi>S</mi> <mi>i</mi></msub><mo>)</mo></mrow></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>γ</mi><msub><mo stretchy="false">)</mo> <mfrac linethickness="0"><mrow><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></mrow><mrow><mi>γ</mi><mo>∈</mo><msub><mi>K</mi> <mi>i</mi></msub></mrow></mfrac></msub></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mi>n</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>n</mi></msub></mrow></mover></mtd> <mtd><mi>F</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>po</mi> <mi>h</mi></msup><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><mo>∃</mo><msub><mi>η</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∃</mo><msub><mi>η</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>epi</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>n</mi></msub></mrow></mover></mtd> <mtd><mi>ker</mi><msub><mrow><mo>(</mo><mo stretchy="false">(</mo><msup><mi>γ</mi> <mo>*</mo></msup><mo>)</mo></mrow> <mfrac linethickness="0"><mrow><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></mrow><mrow><mrow><mi>γ</mi><mo>∈</mo><msub><mi>K</mi> <mi>i</mi></msub></mrow></mrow></mfrac></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>η</mi> <mi>n</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msup><mi>γ</mi> <mo>*</mo></msup><msub><mo stretchy="false">)</mo> <mfrac linethickness="0"><mrow><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></mrow><mrow><mrow><mi>γ</mi><mo>∈</mo><msub><mi>K</mi> <mi>i</mi></msub></mrow></mrow></mfrac></msub></mrow></munder></mtd> <mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mfrac linethickness="0"><mrow><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></mrow><mrow><mrow><mi>γ</mi><mo>∈</mo><msub><mi>K</mi> <mi>i</mi></msub></mrow></mrow></mfrac></munder><mi>F</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \left( \underset{{i \in I}\atop {\gamma \in K_i}}{\sqcup} S_i \right) &\overset{(\gamma)_{{i \in I}\atop \gamma \in K_i}}{\longrightarrow}& X_n &\overset{\eta_n}{\longrightarrow}& F \\ \downarrow &(po^{h})& \downarrow & \nearrow_{\mathrlap{\exists \eta_{n+1}}} \\ \ast &\longrightarrow& X_{n+1} } \;\;\;\;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\;\;\;\; \array{ && F(X_{n+1}) &\longrightarrow& \ast \\ &{}^{\mathllap{\exists \eta_{n+1}}}\nearrow& \downarrow^{\mathrlap{epi}} && \downarrow \\ \ast &\overset{\eta_n}{\longrightarrow}& ker\left((\gamma^\ast\right)_{{i \in I} \atop {\gamma \in K_i}}) &\longrightarrow& \ast \\ &{}_{\mathllap{\eta_n}}\searrow& \downarrow &(pb)& \downarrow \\ && F(X_n) &\underset{(\gamma^\ast)_{{i \in I} \atop {\gamma \in K_i}} }{\longrightarrow}& \underset{{i \in I}\atop {\gamma\in K_i}}{\prod}F(S_i) } \,. </annotation></semantics></math></div> <p>It is now clear that we want to take</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>′</mo><mo>≔</mo><msub><munder><mi>lim</mi><mo>→</mo></munder> <mi>n</mi></msub><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> X' \coloneqq \underset{\rightarrow}{\lim}_n X_n </annotation></semantics></math></div> <p>and extend all the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\eta_n</annotation></semantics></math> to that colimit. Since we have no condition for evaluating <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> on colimits other than pushouts, observe that this <a class="existingWikiWord" href="/nlab/show/sequential+colimit">sequential colimit</a> is equivalent to the following pushout:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo>⊔</mo><mi>n</mi></munder><msub><mi>X</mi> <mi>n</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><munder><mo>⊔</mo><mi>n</mi></munder><msub><mi>X</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>po</mi> <mi>h</mi></msup><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><munder><mo>⊔</mo><mi>n</mi></munder><msub><mi>X</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><mo>′</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \underset{n}{\sqcup} X_n &\longrightarrow& \underset{n}{\sqcup} X_{2n} \\ \downarrow &(po^h)& \downarrow \\ \underset{n}{\sqcup} X_{2n+1} &\longrightarrow& X' } \,, </annotation></semantics></math></div> <p>where the components of the top and left map alternate between the identity on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_n</annotation></semantics></math> and the above successor maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_n \to X_{n+1}</annotation></semantics></math>. Now the excision property of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> applies to this pushout, and we conclude the desired extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>′</mo><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>′</mo><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\eta' \colon X' \to F</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><munder><mo>⊔</mo><mi>n</mi></munder><msub><mi>X</mi> <mi>n</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><munder><mo>⊔</mo><mi>n</mi></munder><msub><mi>X</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><mo>′</mo></mtd> <mtd><mo>⟵</mo></mtd> <mtd><munder><mo>⊔</mo><mi>n</mi></munder><msub><mi>X</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>η</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∃</mo><mi>η</mi><mo>′</mo></mrow></mpadded></msup></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>η</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>F</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo>′</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∃</mo><mi>η</mi><mo>′</mo></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>epi</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>*</mo><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>η</mi> <mi>n</mi></msub><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow></mover></mtd> <mtd><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>n</mi></msub><mi>F</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mi>n</mi></munder><mi>F</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mi>n</mi></munder><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mi>n</mi></munder><mi>F</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ && \underset{n}{\sqcup} X_n \\ & \swarrow && \searrow \\ \underset{n}{\sqcup} X_{2n+1} &\longrightarrow& X' &\longleftarrow& \underset{n}{\sqcup} X_{2n} \\ & {}_{\mathllap{(\eta_{2n+1})_{n}}}\searrow& \downarrow^{\mathrlap{\exists \eta'}} & \swarrow_{\mathrlap{(\eta_{2n})_n}} \\ && F } \;\;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\;\; \array{ && F(X') \\ &{}^{\mathllap{\exists \eta'}}\nearrow& \downarrow^{\mathrlap{epi}} \\ &\ast \overset{(\eta_n)_n}{\longrightarrow}& \underset{\longleftarrow}{\lim}_n F(X_n) \\ & \swarrow && \searrow \\ \underset{n}{\prod}F(X_{2n+1}) && && \underset{n}{\prod}(X_{2n}) \\ & \searrow && \swarrow \\ && \underset{n}{\prod}F(X_n) } \,, </annotation></semantics></math></div> <p>It remains to confirm that this indeed gives the desired bijection. Surjectivity is clear. For injectivity use that all the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">S_i</annotation></semantics></math> are, by assumption, <a class="existingWikiWord" href="/nlab/show/compact+object">compact</a>, hence they may be taken inside the <a class="existingWikiWord" href="/nlab/show/sequential+colimit">sequential colimit</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mrow><mi>n</mi><mo stretchy="false">(</mo><mi>γ</mi><mo stretchy="false">)</mo></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∃</mo><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>S</mi> <mi>i</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mi>γ</mi></mover></mtd> <mtd><mi>X</mi><mo>′</mo><mo>=</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>n</mi></msub><msub><mi>X</mi> <mi>n</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && X_{n(\gamma)} \\ &{}^{\mathllap{ \exists \hat \gamma}}\nearrow& \downarrow \\ S_i &\overset{\gamma}{\longrightarrow}& X' = \underset{\longrightarrow}{\lim}_n X_n } \,. </annotation></semantics></math></div> <p>With this, injectivity follows because by construction we quotiented out the kernel at each stage. Because suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> is taken to zero in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(S_i)</annotation></semantics></math>, then by the definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{n+1}</annotation></semantics></math> above there is a factorization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> through the point:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mn>0</mn><mo lspace="verythinmathspace">:</mo></mtd> <mtd><msub><mi>S</mi> <mi>i</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mover></mtd> <mtd><msub><mi>X</mi> <mrow><mi>n</mi><mo stretchy="false">(</mo><mi>γ</mi><mo stretchy="false">)</mo></mrow></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>n</mi></msub></mrow></mover></mtd> <mtd><mi>F</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mrow><mi>n</mi><mo stretchy="false">(</mo><mi>γ</mi><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>X</mi><mo>′</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ 0 \colon & S_i &\overset{\hat \gamma}{\longrightarrow}& X_{n(\gamma)} &\overset{\eta_n}{\longrightarrow}& F \\ & \downarrow && \downarrow & \\ & \ast &\longrightarrow& X_{n(\gamma)+1} \\ & && \downarrow \\ & && X' } </annotation></semantics></math></div> <p>This concludes the proof of Lemma (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋆</mo></mrow><annotation encoding="application/x-tex">\star</annotation></semantics></math>).</p> <p>Now apply the construction given by this lemma to the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>≔</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">X_0 \coloneqq 0</annotation></semantics></math> and the unique <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mn>0</mn></msub><mo lspace="verythinmathspace">:</mo><mn>0</mn><mover><mo>→</mo><mrow><mo>∃</mo><mo>!</mo></mrow></mover><mi>F</mi></mrow><annotation encoding="application/x-tex">\eta_0 \colon 0 \stackrel{\exists !}{\to} F</annotation></semantics></math>. Lemma <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⋆</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\star)</annotation></semantics></math> then produces an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">X'</annotation></semantics></math> which represents <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> on all the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">S_i</annotation></semantics></math>, and we want to show that this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">X'</annotation></semantics></math> actually represents <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> generally, hence that for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">Y \in \mathcal{C}</annotation></semantics></math> the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>≔</mo><mi>η</mi><mo>′</mo><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo>′</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow></mrow></mover><mi>F</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \theta \coloneqq \eta'\circ (-) \;\colon\; Ho(\mathcal{C})(Y,X') \stackrel{}{\longrightarrow} F(Y) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a>.</p> <p>First, to see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> is surjective, we need to find a preimage of any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo lspace="verythinmathspace">:</mo><mi>Y</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\rho \colon Y \to F</annotation></semantics></math>. Applying Lemma <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⋆</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\star)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>η</mi><mo>′</mo><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>′</mo><mo>⊔</mo><mi>Y</mi><mo>⟶</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">(\eta',\rho)\colon X'\sqcup Y \longrightarrow F</annotation></semantics></math> we get an extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math> of this through some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>′</mo><mo>⊔</mo><mi>Y</mi><mo>⟶</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">X' \sqcup Y \longrightarrow Z</annotation></semantics></math> and the morphism on the right of the following commuting diagram:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>X</mi><mo>′</mo><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>η</mi><mo>′</mo><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><mi>κ</mi><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Ho(\mathcal{C})(-,X') && \longrightarrow && Ho(\mathcal{C})(-, Z) \\ & {}_{\mathllap{\eta'\circ(-)}}\searrow && \swarrow_{\mathrlap{\kappa \circ (-)}} \\ && F(-) } \,. </annotation></semantics></math></div> <p>Moreover, Lemma <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⋆</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\star)</annotation></semantics></math> gives that evaluated on all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">S_i</annotation></semantics></math>, the two diagonal morphisms here become isomorphisms. But then prop. <a class="maruku-ref" href="#WhiteheadTheoremForCompactGenerationByCogroupObjects"></a> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>′</mo><mo>⟶</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">X' \longrightarrow Z</annotation></semantics></math> is in fact an equivalence. Hence the component map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>Z</mi><mo>≃</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">Y \to Z \simeq Z</annotation></semantics></math> is a lift of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math> through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math>.</p> <p>Second, to see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> is injective, suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>Y</mi><mo>→</mo><mi>X</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f,g \colon Y \to X'</annotation></semantics></math> have the same image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math>. Then consider their <a class="existingWikiWord" href="/nlab/show/homotopy+pushout">homotopy pushout</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Y</mi><mo>⊔</mo><mi>Y</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi><mo>′</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Z</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Y \sqcup Y &\stackrel{(f,g)}{\longrightarrow}& X' \\ \downarrow && \downarrow \\ Y &\longrightarrow& Z } </annotation></semantics></math></div> <p>along the <a class="existingWikiWord" href="/nlab/show/codiagonal">codiagonal</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>. Using that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> sends this to a <a class="existingWikiWord" href="/nlab/show/weak+pullback">weak pullback</a> by assumption, we obtain an extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>η</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar \eta</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\eta'</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>′</mo><mo>→</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">X' \to Z</annotation></semantics></math>. Applying Lemma <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⋆</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\star)</annotation></semantics></math> to this gives a further extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>η</mi><mo stretchy="false">¯</mo></mover><mo>′</mo><mo lspace="verythinmathspace">:</mo><mi>Z</mi><mo>′</mo><mo>→</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">\bar \eta' \colon Z' \to Z</annotation></semantics></math> which now makes the following diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>X</mi><mo>′</mo><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>η</mi><mo>′</mo><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><mover><mi>η</mi><mo stretchy="false">¯</mo></mover><mo>′</mo><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Ho(\mathcal{C})(-,X') && \longrightarrow && Ho(\mathcal{C})(-, Z) \\ & {}_{\mathllap{\eta'\circ(-)}}\searrow && \swarrow_{\mathrlap{\bar \eta' \circ (-)}} \\ && F(-) } </annotation></semantics></math></div> <p>such that the diagonal maps become isomorphisms when evaluated on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">S_i</annotation></semantics></math>. As before, it follows via prop. <a class="maruku-ref" href="#WhiteheadTheoremForCompactGenerationByCogroupObjects"></a> that the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>′</mo><mo>⟶</mo><mi>Z</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">h \colon X' \longrightarrow Z'</annotation></semantics></math> is an equivalence.</p> <p>Since by this construction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∘</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">h\circ f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∘</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">h\circ g</annotation></semantics></math> are homotopic</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Y</mi><mo>⊔</mo><mi>Y</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi><mo>′</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mover><mo>≃</mo><mi>h</mi></mover></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Z</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Z</mi><mo>′</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Y \sqcup Y &\stackrel{(f,g)}{\longrightarrow}& X' \\ \downarrow && \downarrow & \searrow^{\mathrlap{\stackrel{h}{\simeq}}} \\ Y &\longrightarrow& Z &\longrightarrow& Z' } </annotation></semantics></math></div> <p>it follows with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> being an equivalence that already <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> were homotopic, hence that they represented the same element.</p> </div> <div class="num_defn" id="GeneralizedCohomologyOnGeneralInfinityCategory"> <h6 id="definition_3">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> with <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pushouts">(∞,1)-pushouts</a>, and with a <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">0 \in \mathcal{C}</annotation></semantics></math>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo lspace="verythinmathspace">:</mo><mi>𝒞</mi><mo>→</mo><mi>𝒞</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>↦</mo><mn>0</mn><munder><mo>⊔</mo><mi>X</mi></munder><mn>0</mn></mrow><annotation encoding="application/x-tex">\Sigma \colon \mathcal{C} \to \mathcal{C}\colon X\mapsto 0 \underset{X}{\sqcup} 0</annotation></semantics></math> for the corresponding <a class="existingWikiWord" href="/nlab/show/suspension">suspension</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a>.</p> <p>A <strong>reduced additive <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> theory</strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>⟶</mo><msup><mi>Ab</mi> <mi>ℤ</mi></msup></mrow><annotation encoding="application/x-tex"> H^\bullet \;\colon \; Ho(\mathcal{C})^{op} \longrightarrow Ab^{\mathbb{Z}} </annotation></semantics></math></div> <p>(from the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite</a> of the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category">homotopy category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+abelian+groups">graded abelian groups</a>);</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/natural+isomorphisms">natural isomorphisms</a> (“<a class="existingWikiWord" href="/nlab/show/suspension+isomorphisms">suspension isomorphisms</a>”) of degree +1</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mo>•</mo></msup><mo>⟶</mo><msup><mi>H</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo>∘</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex"> \delta \; \colon \; H^\bullet \longrightarrow H^{\bullet+1} \circ \Sigma </annotation></semantics></math></div></li> </ol> <p>such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">H^\bullet</annotation></semantics></math></p> <ol> <li> <p><strong>(exactness)</strong> takes <a class="existingWikiWord" href="/nlab/show/homotopy+cofiber+sequences">homotopy cofiber sequences</a> to <a class="existingWikiWord" href="/nlab/show/exact+sequences">exact sequences</a>.</p> </li> <li> <p><strong>(additivity)</strong> takes small <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> to <a class="existingWikiWord" href="/nlab/show/products">products</a>;</p> </li> </ol> </div> <div class="num_defn" id="ConnectinHomomorphismForCohomologyTheoryOnInfinityCategory"> <h6 id="definition_4">Definition</h6> <p>Given a generalized cohomology theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>H</mi> <mo>•</mo></msup><mo>,</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(H^\bullet,\delta)</annotation></semantics></math> on some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> as in def. <a class="maruku-ref" href="#GeneralizedCohomologyOnGeneralInfinityCategory"></a>, and given a <a class="existingWikiWord" href="/nlab/show/homotopy+cofiber+sequence">homotopy cofiber sequence</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>Y</mi><mover><mo>⟶</mo><mi>g</mi></mover><mi>Z</mi><mover><mo>⟶</mo><mrow><mi>coker</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mover><mi>Σ</mi><mi>X</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> X \stackrel{f}{\longrightarrow} Y \stackrel{g}{\longrightarrow} Z \stackrel{coker(g)}{\longrightarrow} \Sigma X \,, </annotation></semantics></math></div> <p>then the corresponding <strong><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></strong> is the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>δ</mi></mover><msup><mi>H</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>coker</mi><mo stretchy="false">(</mo><mi>g</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow></mover><msup><mi>H</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \partial \;\colon\; H^\bullet(X) \stackrel{\delta}{\longrightarrow} H^{\bullet+1}(\Sigma X) \stackrel{coker(g)^\ast}{\longrightarrow} H^{\bullet+1}(Z) \,. </annotation></semantics></math></div></div> <div class="num_prop" id="LongExactSequenceOfACohomologyTheoryOnAnInfinityCategory"> <h6 id="proposition_2">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/connecting+homomorphisms">connecting homomorphisms</a> of def. <a class="maruku-ref" href="#ConnectinHomomorphismForCohomologyTheoryOnInfinityCategory"></a> are parts of <a class="existingWikiWord" href="/nlab/show/long+exact+sequences">long exact sequences</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mover><mo>⟶</mo><mo>∂</mo></mover><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>∂</mo></mover><msup><mi>H</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \stackrel{\partial}{\longrightarrow} H^{\bullet}(Z) \longrightarrow H^\bullet(Y) \longrightarrow H^\bullet(X) \stackrel{\partial}{\longrightarrow} H^{\bullet+1}(Z) \to \cdots \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>By the defining exactness of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">H^\bullet</annotation></semantics></math>, def. <a class="maruku-ref" href="#GeneralizedCohomologyOnGeneralInfinityCategory"></a>, and the way this appears in def. <a class="maruku-ref" href="#ConnectinHomomorphismForCohomologyTheoryOnInfinityCategory"></a>, using that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math> is by definition an isomorphism.</p> </div> <div class="num_prop" id="CohomologyFunctorOnInfinityCategoryIsBrownFunctor"> <h6 id="proposition_3">Proposition</h6> <p>Given a reduced addive generalized cohomology functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo lspace="verythinmathspace">:</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><msup><mi>Ab</mi> <mi>ℤ</mi></msup></mrow><annotation encoding="application/x-tex">H^\bullet \colon Ho(\mathcal{C})^{op}\to Ab^{\mathbb{Z}}</annotation></semantics></math>, def. <a class="maruku-ref" href="#GeneralizedCohomologyOnGeneralInfinityCategory"></a>, its underlying <a class="existingWikiWord" href="/nlab/show/Set">Set</a>-valued functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>n</mi></msup><mo lspace="verythinmathspace">:</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><mi>Ab</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">H^n \colon Ho(\mathcal{C})^{op}\to Ab\to Set</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/Brown+functors">Brown functors</a>, def. <a class="maruku-ref" href="#BrownFunctorOnInfinityCategory"></a>.</p> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>The first condition on a <a class="existingWikiWord" href="/nlab/show/Brown+functor">Brown functor</a> holds by additivity. For the second condition, given a homotopy pushout square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msub><mi>Y</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mrow></mrow></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msub><mi>Y</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X_1 &\stackrel{f_1}{\longrightarrow}& Y_1 \\ \downarrow^{} && \downarrow \\ X_2 &\stackrel{f_2}{\longrightarrow}& Y_2 } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, consider the induced morphism of the <a class="existingWikiWord" href="/nlab/show/long+exact+sequences">long exact sequences</a> given by prop. <a class="maruku-ref" href="#LongExactSequenceOfACohomologyTheoryOnAnInfinityCategory"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>coker</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mi>Y</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msubsup><mi>f</mi> <mn>2</mn> <mo>*</mo></msubsup></mrow></mover></mtd> <mtd><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><msup><mi>H</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Σ</mi><mi>coker</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>coker</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mi>Y</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msubsup><mi>f</mi> <mn>1</mn> <mo>*</mo></msubsup></mrow></mover></mtd> <mtd><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><msup><mi>H</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Σ</mi><mi>coker</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ H^\bullet(coker(f_2)) &\longrightarrow& H^\bullet(Y_2) &\stackrel{f^\ast_2}{\longrightarrow}& H^\bullet(X_2) &\stackrel{}{\longrightarrow}& H^{\bullet+1}(\Sigma coker(f_2)) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}} \\ H^\bullet(coker(f_1)) &\longrightarrow& H^\bullet(Y_1) &\stackrel{f^\ast_1}{\longrightarrow}& H^\bullet(X_1) &\stackrel{}{\longrightarrow}& H^{\bullet+1}(\Sigma coker(f_1)) } </annotation></semantics></math></div> <p>Here the outer vertical morphisms are <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>, as shown, due to the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a> (see also at <em><a href="homotopy+pullback#FiberwiseRecognitionInStableCase">fiberwise recognition of stable homotopy pushouts</a></em>). This means that the <a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a> applies to this diagram. Inspection shows that this implies the claim.</p> </div> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> which satisfies the conditions of theorem <a class="maruku-ref" href="#BrownRepresentabilityOnPresentableInfinityCategories"></a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>H</mi> <mo>•</mo></msup><mo>,</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(H^\bullet, \delta)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/generalized+cohomology">generalized cohomology</a> functor on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, def. <a class="maruku-ref" href="#GeneralizedCohomologyOnGeneralInfinityCategory"></a>. Then there exists a <a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>∈</mo><mi>Stab</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E \in Stab(\mathcal{C})</annotation></semantics></math> such that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">H^\bullet</annotation></semantics></math> is degreewise <a class="existingWikiWord" href="/nlab/show/representable+functor">represented</a> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo>≃</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msub><mi>E</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> H^\bullet \simeq Ho(\mathcal{C})(-,E_\bullet) \,, </annotation></semantics></math></div></li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/suspension+isomorphism">suspension isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math> is given by the structure morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>σ</mi><mo stretchy="false">˜</mo></mover> <mi>n</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>E</mi> <mi>n</mi></msub><mo>→</mo><mi>Ω</mi><msub><mi>E</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\tilde \sigma_n \colon E_n \to \Omega E_{n+1}</annotation></semantics></math> of the spectrum, in that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo lspace="verythinmathspace">:</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msub><mi>E</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msub><mover><mi>σ</mi><mo stretchy="false">˜</mo></mover> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow></mover><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>Ω</mi><msub><mi>E</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>E</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \delta \colon H^n(-) \simeq Ho(\mathcal{C})(-,E_n) \stackrel{Ho(\mathcal{C})(-,\tilde\sigma_n) }{\longrightarrow} Ho(\mathcal{C})(-,\Omega E_{n+1}) \simeq Ho(\mathcal{C})(\Sigma (-), E_{n+1}) \simeq H^{n+1}(\Sigma(-)) \,. </annotation></semantics></math></div></li> </ol> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>Via prop. <a class="maruku-ref" href="#CohomologyFunctorOnInfinityCategoryIsBrownFunctor"></a>, theorem <a class="maruku-ref" href="#BrownRepresentabilityOnPresentableInfinityCategories"></a> gives the first clause. With this, the second clause follows by the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a> (in fact just with the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>).</p> </div> <h2 id="further_variants">Further variants</h2> <h3 id="for_triangulated_categories_and_model_categories">For triangulated categories and model categories</h3> <p>For <a class="existingWikiWord" href="/nlab/show/triangulated+categories">triangulated categories</a> a discussion is in (<a href="#Neeman96">Neeman 96</a>).</p> <p>For <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> a discussion is in (<a href="#Jardine09">Jardine 09</a>):</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> with a <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a> and such that there is a <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/cofibrant+object">cofibrant</a> <a class="existingWikiWord" href="/nlab/show/compact+objects">compact objects</a> such that the <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> are precisely the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/local+morphisms">local morphisms</a>.</p> <p>(<a href="#Jardine09">Jardine 09, p. 20</a>)</p> <div class="num_theorem"> <h6 id="theorem_3">Theorem</h6> <p><strong>(Jardine)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><msub><mi>Set</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">F : C^{op} \to Set_{*}</annotation></semantics></math> be a pointed <a class="existingWikiWord" href="/nlab/show/homotopical+functor">homotopical functor</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">C^{op}</annotation></semantics></math> to the category of <a class="existingWikiWord" href="/nlab/show/pointed+sets">pointed sets</a> such that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> preserves small <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> of cofibrant objects (including preserving the <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a>).</p> </li> <li> <p><strong>(Mayer-Vietoris property)</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> takes any <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mi>i</mi></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>B</mi><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>A</mi></munder><mi>X</mi><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\to& X \\ \downarrow^i && \downarrow \\ B &\to& B \coprod_A X, } </annotation></semantics></math></div> <p>with all objects cofibrant and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> a cofibration, to a <a class="existingWikiWord" href="/nlab/show/weak+pullback">weak pullback</a>.</p> </li> </ol> <p>(<a href="#Jardine09">Jardine 09, p. 21</a>)</p> <p>Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a>.</p> </div> <p>(<a href="#Jardine09">Jardine 09, theorem 24</a>)</p> <h3 id="ForEquivariantCohomology">For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RO</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">RO(G)</annotation></semantics></math>-graded equivariant cohomology</h3> <p>The generalization of the Brown representability theorem to <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a> says that <a class="existingWikiWord" href="/nlab/show/RO%28G%29-grading">RO(G)-graded</a> equivariant cohomology is represented by <a class="existingWikiWord" href="/nlab/show/genuine+G-spectra">genuine G-spectra</a> (e.g. <a href="#May96">May 96, chapter XIII.3</a>).</p> <h2 id="properties">Properties</h2> <h3 id="multiplicative_cohomology_and_ring_spectra">Multiplicative cohomology and ring spectra</h3> <p>The spectrum Brown-representing a <a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a> inherits (at least) the structure of an <a class="existingWikiWord" href="/nlab/show/H-ring+spectrum">H-ring spectrum</a>. See <a href="multiplicative+cohomology+theory#BrownRepresentabilityByRingSpectra">there</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Landweber+exact+functor+theorem">Landweber exact functor theorem</a></li> </ul> <h2 id="references">References</h2> <p>The original version of the theorem:</p> <ul> <li id="Brown62"><a class="existingWikiWord" href="/nlab/show/Edgar+Brown">Edgar Brown</a>, <em>Cohomology theories</em>, Annals of Mathematics, Second Series 75: 467–484 (1962) (<a href="https://www.jstor.org/stable/1970209">jstor:1970209</a>)</li> </ul> <p>The category-theoretic generalization:</p> <ul> <li id="Brown65"><a class="existingWikiWord" href="/nlab/show/Edgar+Brown">Edgar Brown</a>, <em>Abstract homotopy theory</em>, Trans. AMS 119 no. 1 (1965) (<a href="https://doi.org/10.1090/S0002-9947-1965-0182970-6">doi:10.1090/S0002-9947-1965-0182970-6</a>)</li> </ul> <p>A simplified version of the proof was spelled out in</p> <ul> <li id="Spanier66"><a class="existingWikiWord" href="/nlab/show/Edwin+Spanier">Edwin Spanier</a>, section 7.7 of <em>Algebraic topology</em>, McGraw-Hill, 1966</li> </ul> <p>and a strengthening in</p> <ul> <li id="Adams71"><a class="existingWikiWord" href="/nlab/show/John+Frank+Adams">John Frank Adams</a>, <em>A variant of E. H. Brown’s representability theorem</em>, Topology, 10:185-198, 1971</li> </ul> <p>Textbook accounts:</p> <ul> <li id="Adams74"> <p><a class="existingWikiWord" href="/nlab/show/Frank+Adams">Frank Adams</a>, part III, section 6 of <em><a class="existingWikiWord" href="/nlab/show/Stable+homotopy+and+generalised+homology">Stable homotopy and generalised homology</a></em>, 1974</p> </li> <li id="Switzer75"> <p><a class="existingWikiWord" href="/nlab/show/Robert+Switzer">Robert Switzer</a>, theorem 8.42, theorem 9.27 in <em>Algebraic Topology - Homotopy and Homology</em>, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.</p> </li> <li id="Kochmann96"> <p><a class="existingWikiWord" href="/nlab/show/Stanley+Kochmann">Stanley Kochmann</a>, section 3.4 of <em><a class="existingWikiWord" href="/nlab/show/Bordism%2C+Stable+Homotopy+and+Adams+Spectral+Sequences">Bordism, Stable Homotopy and Adams Spectral Sequences</a></em>, AMS 1996</p> </li> <li id="AguilarGitlerPrito02"> <p>Marcelo Aguilar, <a class="existingWikiWord" href="/nlab/show/Samuel+Gitler">Samuel Gitler</a>, Carlos Prieto, section 12 of <em>Algebraic topology from a homotopical viewpoint</em>, Springer (2002) (<a href="http://tocs.ulb.tu-darmstadt.de/106999419.pdf">toc pdf</a>)</p> </li> <li id="TamakiKono06"> <p><a class="existingWikiWord" href="/nlab/show/Dai+Tamaki">Dai Tamaki</a>, <a class="existingWikiWord" href="/nlab/show/Akira+Kono">Akira Kono</a>, Sections 2.4, 2.5 in: <em>Generalized Cohomology</em>, Translations of Mathematical Monographs, American Mathematical Society, 2006 (<a href="https://bookstore.ams.org/mmono-230">ISBN: 978-0-8218-3514-2</a>)</p> </li> </ul> <p>Quick reviews include</p> <ul> <li id="Muro10"> <p><a class="existingWikiWord" href="/nlab/show/Fernando+Muro">Fernando Muro</a>, <em>Representability of cohomology theories</em>, 2010 (<a href="http://personal.us.es/fmuro/praha.pdf">pdf</a>)</p> </li> <li id="Lurie"> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Chromatic+Homotopy+Theory">Chromatic Homotopy Theory</a></em>, Lecture series 2010, Lecture 17 <em>Phantom maps</em> (<a href="http://www.math.harvard.edu/~lurie/252xnotes/Lecture17.pdf">pdf</a>)</p> </li> </ul> <p>Generalization to <a class="existingWikiWord" href="/nlab/show/triangulated+categories">triangulated categories</a> is discussed in</p> <ul> <li id="Neeman96"><a class="existingWikiWord" href="/nlab/show/Amnon+Neeman">Amnon Neeman</a>, <em>The Grothendieck duality theorem via Bousfield’s techniques and Brown representability</em>, J. Amer. Math. Soc. 9 (1996), 205-236 (<a href="http://www.ams.org/journals/jams/1996-9-01/S0894-0347-96-00174-9/">AMS</a>)</li> </ul> <p>A <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> version, with applications to <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stacks">∞-stacks</a> – or rather to the standard <a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a> in terms of the standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> – is given in</p> <ul> <li id="Jardine09"><a class="existingWikiWord" href="/nlab/show/Rick+Jardine">Rick Jardine</a>, <em>Representability theorems for simplicial</em> <p>presheaves_, 2009 (<a href="http://ncatlab.org/nlab/files/JardineBrownrep.pdf">pdf</a>)</p> </li> </ul> <p>A version in <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category+theory">(infinity,1)-category theory</a> is in</p> <ul> <li id="LurieHigherAlgebra"><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, section 1.4.1 of <em><a class="existingWikiWord" href="/nlab/show/Higher+Algebra">Higher Algebra</a></em></li> </ul> <p>Exposition of this is in</p> <ul> <li id="Mathew11"><a class="existingWikiWord" href="/nlab/show/Akhil+Mathew">Akhil Mathew</a>, <em><a href="https://amathew.wordpress.com/2011/10/10/brown-representability-and-infinity-categories/#more-2907">Brown representability and infinity-categories</a></em>, 2011</li> </ul> <p>The case of <a class="existingWikiWord" href="/nlab/show/RO%28G%29-grading">RO(G)-graded</a> <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a> theory is discussed for instance in</p> <ul> <li id="May96"><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, chapter XIII.3 of <em>Equivariant homotopy and cohomology theory</em>, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. (<a href="http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf">pdf</a>)</li> </ul> <p>The counterexamples to nonconnected and unpointed Brown representability are from</p> <ul> <li id="FreydHeller93"><a class="existingWikiWord" href="/nlab/show/Peter+Freyd">Peter Freyd</a>, <a class="existingWikiWord" href="/nlab/show/Alex+Heller">Alex Heller</a>, <em>Splitting homotopy idempotents. II.</em> J. Pure App#l. Algebra 89 (1993), no. 1-2, 93–106.</li> </ul> <p>The relation to <a class="existingWikiWord" href="/nlab/show/Grothendieck+contexts">Grothendieck contexts</a> (<a class="existingWikiWord" href="/nlab/show/six+operations">six operations</a>) is highlighted in</p> <ul> <li id="Neeman96"><a class="existingWikiWord" href="/nlab/show/Amnon+Neeman">Amnon Neeman</a>, <em>The Grothendieck duality theorem via Bousfield’s techniques and Brown representability</em>, J. Amer. Math. Soc. 9 (1996), 205-236 (<a href="http://www.ams.org/journals/jams/1996-9-01/S0894-0347-96-00174-9/">web</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 26, 2021 at 15:56:59. 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