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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> (infinity,1)-topos (changes) </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/diff/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/9467/#Item_9" 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" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #88 to #89: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='category_theory'><math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_1' xmlns='http://www.w3.org/1998/Math/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 theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a></strong></p> <p><strong>Background</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28n%2Cr%29-category'>(n,r)-category</a></p> </li> </ul> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hom-object+in+a+quasi-category'>hom-objects</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equivalence+in+an+%28infinity%2C1%29-category'>equivalences in</a>/<a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+%28infinity%2C1%29-categories'>of</a> <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_2' xmlns='http://www.w3.org/1998/Math/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</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sub-%28infinity%2C1%29-category'>sub-(∞,1)-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/reflective+sub-%28infinity%2C1%29-category'>reflective sub-(∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/localization+of+an+%28infinity%2C1%29-category'>reflective localization</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/opposite+quasi-category'>opposite (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/over-%28infinity%2C1%29-category'>over (∞,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/join+of+quasi-categories'>join of quasi-categories</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functor</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/flat+%28infinity%2C1%29-functor'>exact (∞,1)-functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-functors'>(∞,1)-category of (∞,1)-functors</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-presheaves'>(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> </li> <li> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/fibration+of+quasi-categories'>fibrations</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inner+fibration'>inner fibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/right%2Fleft+Kan+fibration'>left/right fibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cartesian+fibration'>Cartesian fibration</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Cartesian+morphism'>Cartesian morphism</a></li> </ul> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>limit</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/terminal+object+in+a+quasi-category'>terminal object</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>adjoint functors</a></p> </li> </ul> <p><strong>Local presentation</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>locally presentable</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/essentially+small+%28infinity%2C1%29-category'>essentially small</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+small+%28infinity%2C1%29-category'>locally small</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/accessible+%28infinity%2C1%29-category'>accessible</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/idempotent+complete+%28infinity%2C1%29-category'>idempotent-complete</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories'>(∞,1)-Yoneda lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-Grothendieck+construction'>(∞,1)-Grothendieck construction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor+theorem'>adjoint (∞,1)-functor theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monadicity+theorem'>(∞,1)-monadicity theorem</a></p> </li> </ul> <p><strong>Extra stuff, structure, properties</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category'>stable (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category+with+weak+equivalences'>category with weak equivalences</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/derivator'>derivator</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quasi-category'>quasi-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+quasi-categories'>model structure for quasi-categories</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+Cartesian+fibrations'>model structure for Cartesian fibrations</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/relation+between+quasi-categories+and+simplicial+categories'>relation to simplicial categories</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coherent+nerve'>homotopy coherent nerve</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simplicial+model+category'>simplicial model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presentable quasi-category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+sets'>model structure for Kan complexes</a></li> </ul> </li> </ul> </div> <h4 id='topos_theory'><math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_3' xmlns='http://www.w3.org/1998/Math/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'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%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/diff/sheaf+and+topos+theory'>sheaf and topos theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-presheaf'>(∞,1)-presheaf</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%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/diff/elementary+%28infinity%2C1%29-topos'>elementary (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-site'>(∞,1)-site</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/reflective+sub-%28infinity%2C1%29-category'>reflective sub-(∞,1)-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/localization+of+an+%28infinity%2C1%29-category'>localization of an (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+localization'>topological localization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hypercompletion'>hypercompletion</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-category of (∞,1)-sheaves</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-sheaf'>(∞,1)-sheaf</a>/<a class='existingWikiWord' href='/nlab/show/diff/infinity-stack'>∞-stack</a>/<a class='existingWikiWord' href='/nlab/show/diff/derived+stack'>derived stack</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28n%2C1%29-topos'>(n,1)-topos</a>, <a class='existingWikiWord' href='/nlab/show/diff/n-topos'>n-topos</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/truncation'>n-truncated object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/connected+object'>n-connected object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topos'>(1,1)-topos</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/presheaf'>presheaf</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sheaf'>sheaf</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-topos'>(2,1)-topos</a>, <a class='existingWikiWord' href='/nlab/show/diff/2-topos'>2-topos</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/%282%2C1%29-presheaf'>(2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-quasitopos'>(∞,1)-quasitopos</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/separated+%28infinity%2C1%29-presheaf'>separated (∞,1)-presheaf</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quasitopos'>quasitopos</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/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/diff/separated+%282%2C1%29-presheaf'>separated (2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C2%29-topos'>(∞,2)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2Cn%29-topos'>(∞,n)-topos</a></p> </li> </ul> <h2 id='sidebar_characterization'>Characterization</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pullback-stable+colimit'>universal colimits</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos'>object classifier</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/groupoid+object+in+an+%28infinity%2C1%29-category'>groupoid object in an (∞,1)-topos</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/effective+epimorphism'>effective epimorphism</a></li> </ul> </li> </ul> <h2 id='sidebar_morphisms'>Morphisms</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-geometric+morphism'>(∞,1)-geometric morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29Topos'>(∞,1)Topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/hypercomplete+%28infinity%2C1%29-topos'>hypercomplete (∞,1)-topos</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hypercomplete+object'>hypercomplete object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead theorem</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/over-%28infinity%2C1%29-topos'>over-(∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/n-localic+%28infinity%2C1%29-topos'>n-localic (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+n-connected+%28n%2B1%2C1%29-topos'>locally n-connected (n,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/structured+%28infinity%2C1%29-topos'>structured (∞,1)-topos</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/geometry+%28for+structured+%28infinity%2C1%29-toposes%29'>geometry (for structured (∞,1)-toposes)</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+n-connected+%28n%2B1%2C1%29-topos'>locally ∞-connected (∞,1)-topos</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+n-connected+%28n%2B1%2C1%29-topos'>∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-local+geometric+morphism'>local (∞,1)-topos</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/concrete+%28infinity%2C1%29-sheaf'>concrete (∞,1)-sheaf</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cohesive+%28infinity%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/diff/presentations+of+%28infinity%2C1%29-sheaf+%28infinity%2C1%29-toposes'>models for ∞-stack (∞,1)-toposes</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+functors'>model structure on functors</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+site'>model site</a>/<a class='existingWikiWord' href='/nlab/show/diff/sSet-site'>sSet-site</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+presheaves'>model structure on simplicial presheaves</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/descent+for+simplicial+presheaves'>descent for simplicial presheaves</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/cohesive+%28infinity%2C1%29-topos'>cohesive (∞,1)-topos</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/shape+of+an+%28infinity%2C1%29-topos'>shape</a> / <a class='existingWikiWord' href='/nlab/show/diff/coshape+of+an+%28infinity%2C1%29-topos'>coshape</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+groups+in+an+%28infinity%2C1%29-topos'>homotopy</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid+in+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a>/<a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid+of+a+locally+infinity-connected+%28infinity%2C1%29-topos'>of a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/categorical+homotopy+groups+in+an+%28infinity%2C1%29-topos'>categorical</a>/<a class='existingWikiWord' href='/nlab/show/diff/geometric+homotopy+groups+in+an+%28infinity%2C1%29-topos'>geometric</a> homotopy groups</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Postnikov+tower+in+an+%28infinity%2C1%29-category'>Postnikov tower</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+tower+in+an+%28infinity%2C1%29-topos'>Whitehead tower</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/function+algebras+on+infinity-stacks'>rational homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/dimension'>dimension</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+dimension'>homotopy dimension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cohomological+dimension'>cohomological dimension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/covering+dimension'>covering dimension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#Definition'>Definition</a><ul><li><a href='#AsAGeometricEmbedding'>As a geometric embedding into a <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_4' xmlns='http://www.w3.org/1998/Math/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>-presheaf category</a></li><li><a href='#GiraudAxioms'>By Giraud-Rezk-Lurie axioms</a></li><li><a href='#morphisms_2'>Morphisms</a></li></ul></li><li><a href='#types_of_toposes'>Types of <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_5' xmlns='http://www.w3.org/1998/Math/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>-toposes</a><ul><li><a href='#topological_localizations__sheaf_toposes'>Topological localizations / <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_6' xmlns='http://www.w3.org/1998/Math/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>-sheaf toposes</a></li><li><a href='#hypercomplete_toposes'>Hypercomplete <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_7' xmlns='http://www.w3.org/1998/Math/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>-toposes</a></li><li><a href='#cubical_type_theory'>Cubical type theory</a></li></ul></li><li><a href='#models_2'>Models</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#GlobalSectionsGeometricMorphism'>Global sections geometric morphism</a></li><li><a href='#ClosedMonoidalStructure'>Closed monoidal structure</a></li><li><a href='#PoweringOfInfinityToposesOverInfinityGroupoids'>Powering of <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-toposes over <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoids</a></li><li><a href='#slicetoposes'>Slice-<math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_10' xmlns='http://www.w3.org/1998/Math/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>-toposes</a></li><li><a href='#syntax_in_univalent_homotopy_type_theory'>Syntax in univalent homotopy type theory</a></li></ul></li><li><a href='#ToposTheory'><math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_11' xmlns='http://www.w3.org/1998/Math/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</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a><ul><li><a href='#ReferencesGeneral'>General</a></li><li><a href='#giraudrezklurie_axioms'>Giraud-Rezk-Lurie axioms</a></li><li><a href='#homotopy_type_theory'>Homotopy type theory</a></li></ul></li></ul></div> <h2 id='idea'>Idea</h2> <p>Recall the following familiar 1-categorical statement:</p> <ul> <li>Working in the 1-<a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a> of <a class='existingWikiWord' href='/nlab/show/diff/0-category'>0-categories</a> amounts to doing <a class='existingWikiWord' href='/nlab/show/diff/set+theory'>set theory</a>. The point of <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+topos'>sheaf</a> <a class='existingWikiWord' href='/nlab/show/diff/topos'>toposes</a> is to pass to <em>parameterized</em> <a class='existingWikiWord' href='/nlab/show/diff/0-category'>0-categories</a>, namely <a class='existingWikiWord' href='/nlab/show/diff/presheaf'>presheaf</a> categories. Although these <a class='existingWikiWord' href='/nlab/show/diff/topos'>topoi</a> behave much like the 1-topos <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a>, their objects are generalized <a class='existingWikiWord' href='/nlab/show/diff/space'>spaces</a> that may carry more structure. For instance, a (pre)<a class='existingWikiWord' href='/nlab/show/diff/sheaf'>sheaf</a> on <a class='existingWikiWord' href='/nlab/show/diff/Diff'>Diff</a> is a <a class='existingWikiWord' href='/nlab/show/diff/generalized+smooth+space'>generalized smooth space</a>.</li> </ul> <p>The idea of <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_12' xmlns='http://www.w3.org/1998/Math/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>-toposes is to generalize the above situation from <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_14' xmlns='http://www.w3.org/1998/Math/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> (recall the notion of <a class='existingWikiWord' href='/nlab/show/diff/%28n%2Cr%29-category'>(n,r)-category</a> and see the general discussion at <a class='existingWikiWord' href='/nlab/show/diff/infinity-topos'>∞-topos</a>):</p> <ul> <li>Working in the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a> <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>∞Grpd</a> of <a class='existingWikiWord' href='/nlab/show/diff/infinity-groupoid'>(∞,0)-categories</a> amounts to doing <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>. The point of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-sheaf'>(∞,1)-sheaves</a> is to pass to <em>parameterized</em> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C0%29-category'>(∞,0)-categories</a>, namely <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-presheaf'>(∞,1)-presheaf</a> categories. Although these <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_15' xmlns='http://www.w3.org/1998/Math/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>-topoi behave much like the <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_16' xmlns='http://www.w3.org/1998/Math/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 <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>∞Grpd</a>, their objects are generalized <a class='existingWikiWord' href='/nlab/show/diff/space'>spaces</a> with higher <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopies</a> that may carry more structure. More generally we have topoi of <a class='existingWikiWord' href='/nlab/show/diff/sheaf'>sheaves</a>, and <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_17' xmlns='http://www.w3.org/1998/Math/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>-topoi of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-sheaf'>(∞,1)-sheaves</a>. For instance, an <a class='existingWikiWord' href='/nlab/show/diff/Lie+n-groupoid'>∞-Lie groupoid</a> is an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-sheaf'>(∞,1)-sheaf</a> on <a class='existingWikiWord' href='/nlab/show/diff/CartSp'>CartSp</a>.</li> </ul> <h2 id='Definition'>Definition</h2> <h3 id='AsAGeometricEmbedding'>As a geometric embedding into a <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_18' xmlns='http://www.w3.org/1998/Math/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>-presheaf category</h3> <p>Recall that <a class='existingWikiWord' href='/nlab/show/diff/sheaf+toposes+are+equivalently+the+left+exact+reflective+subcategories+of+presheaf+toposes'>sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes</a> and that the inclusion functor is necessarily an <a class='existingWikiWord' href='/nlab/show/diff/accessible+functor'>accessible functor</a>. This characterization has the following immediate generalization to a definition in <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a>, where the only subtlety is that accessibility needs to be explicitly required:</p> <div class='num_defn' id='ToposByLocalization'> <h6 id='definition_2'>Definition</h6> <p>A <a class='existingWikiWord' href='/nlab/show/diff/Alexander+Grothendieck'>Grothendieck</a>–<a class='existingWikiWord' href='/nlab/show/diff/Charles+Rezk'>Rezk</a>–<a class='existingWikiWord' href='/nlab/show/diff/Jacob+Lurie'>Lurie</a> <strong><math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_19' xmlns='http://www.w3.org/1998/Math/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</strong> <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> is an <a href='reflective%20sub-%28infinity,1%29-category#AccessibleReflectiveSubcategory'>accessible</a> <a href='reflective+sub-%28infinity%2C1%29-category#ExactLocalizations'>left exact</a> <a class='existingWikiWord' href='/nlab/show/diff/reflective+sub-%28infinity%2C1%29-category'>reflective sub-(∞,1)-category</a> of an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-presheaves'>(∞,1)-category of (∞,1)-presheaves</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle><mover><mo>↪</mo><mover><mo>←</mo><mi>lex</mi></mover></mover><msub><mi>PSh</mi> <mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \mathbf{H} \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow} PSh_{(\infty,1)}(C) \,. </annotation></semantics></math></div> <p>If the above localization is a <a class='existingWikiWord' href='/nlab/show/diff/topological+localization'>topological localization</a> then <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> is an <strong><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-category of (∞,1)-sheaves</a></strong>.</p> </div> <h3 id='GiraudAxioms'>By Giraud-Rezk-Lurie axioms</h3> <p>Equivalently:</p> <p>\begin{proposition} <strong>(Giraud-Rezk-Lurie axioms)</strong> \linebreak An <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_23' xmlns='http://www.w3.org/1998/Math/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 <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> is</p> <p>an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a> that satisfies the <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_25' xmlns='http://www.w3.org/1998/Math/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-theoretic analogs of <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+topos'>Giraud's axioms</a>:</p> <ul> <li> <p><math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presentable</a>;</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>(∞,1)-colimits</a> in <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/pullback-stable+colimit'>are universal</a>;</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproducts</a> in <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> are <a class='existingWikiWord' href='/nlab/show/diff/disjoint+coproduct'>disjoint</a>;</p> </li> <li> <p>every <a class='existingWikiWord' href='/nlab/show/diff/groupoid+object+in+an+%28infinity%2C1%29-category'>groupoid object</a> in <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/groupoid+objects+in+an+%28%E2%88%9E%2C1%29-topos+are+effective'>effective</a> (i.e. has a <a class='existingWikiWord' href='/nlab/show/diff/delooping'>delooping</a>).</p> </li> </ul> <p>\end{proposition}</p> <p>This is part of the statement of <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>HTT, theorem 6.1.0.6</a>.</p> <p>This is derived from the following equivalent one:</p> <div class='num_prop' id='CharacterizationByObjectClassifier'> <h6 id='proposition'>Proposition</h6> <p>An <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a> is</p> <ul> <li> <p>a <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presentable (∞,1)-category</a> with <a class='existingWikiWord' href='/nlab/show/diff/pullback-stable+colimit'>universal colimits</a></p> </li> <li> <p>that has <a class='existingWikiWord' href='/nlab/show/diff/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos'>object classifiers</a>.</p> </li> </ul> </div> <div class='num_remark' id='ReflectonOnCharacterizationByObjectClassifier'> <h6 id='remark'>Remark</h6> <p>An <a class='existingWikiWord' href='/nlab/show/diff/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos'>object classifier</a> is a (small) <em>self-reflection</em> of the <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-topos inside itself (<a class='existingWikiWord' href='/nlab/show/diff/type+universe'>type of types</a>, internal <a class='existingWikiWord' href='/nlab/show/diff/universe'>universe</a>). See also (<a href='Science+of+Logic#WesenAlsReflexionInIhmSelbst'>WdL, book 2, section 1</a>).</p> </div> <p>A further equivalent one (essentially by an invocation of the <a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor+theorem'>adjoint functor theorem</a>) is:</p> <div class='num_prop'> <h6 id='proposition_2'>Proposition</h6> <p>An <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a> is</p> <ul> <li> <p>a <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presentable (∞,1)-category</a></p> </li> <li> <p>in which all colimits are <a class='existingWikiWord' href='/nlab/show/diff/van+Kampen+colimit'>van Kampen colimits</a>.</p> </li> </ul> </div> <h3 id='morphisms_2'>Morphisms</h3> <p>A <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphism</a> between <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_31' xmlns='http://www.w3.org/1998/Math/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>-toposes is an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-geometric+morphism'>(∞,1)-geometric morphism</a>.</p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a> of all <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_32' xmlns='http://www.w3.org/1998/Math/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 is <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29Topos'>(∞,1)Toposes</a>.</p> <h2 id='types_of_toposes'>Types of <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_33' xmlns='http://www.w3.org/1998/Math/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>-toposes</h2> <h3 id='topological_localizations__sheaf_toposes'>Topological localizations / <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_34' xmlns='http://www.w3.org/1998/Math/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>-sheaf toposes</h3> <p>for the moment see</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/topological+localization'>topological localization</a></li> </ul> <h3 id='hypercomplete_toposes'>Hypercomplete <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_35' xmlns='http://www.w3.org/1998/Math/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>-toposes</h3> <p>for the moment see</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/hypercomplete+%28infinity%2C1%29-topos'>hypercomplete (∞,1)-topos</a></li> </ul> <h3 id='cubical_type_theory'>Cubical type theory</h3> <p>The Cartesian cubical model of <a class='existingWikiWord' href='/nlab/show/diff/cubical+type+theory'>cubical type theory</a> and <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a> is <a href='https://groups.google.com/d/msg/homotopytypetheory/RQkLWZ_83kQ/s6iazlFdBgAJ'>conjectured</a> to be an (∞,1)-topos not equivalent to (∞,1)-groupoids.</p> <h2 id='models_2'>Models</h2> <p>Another main theorem about <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_36' xmlns='http://www.w3.org/1998/Math/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>-toposes is that <a class='existingWikiWord' href='/nlab/show/diff/presentations+of+%28infinity%2C1%29-sheaf+%28infinity%2C1%29-toposes'>models for ∞-stack (∞,1)-toposes</a> are given by the <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+presheaves'>model structure on simplicial presheaves</a>. See there for details</p> <h2 id='properties'>Properties</h2> <h3 id='GlobalSectionsGeometricMorphism'>Global sections geometric morphism</h3> <p>Every <a class='existingWikiWord' href='/nlab/show/diff/infinity-stack'>∞-stack</a> <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_37' xmlns='http://www.w3.org/1998/Math/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 <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> has a canonical <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-geometric+morphism'>(∞,1)-geometric morphism</a> to the terminal <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-stack <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_40' xmlns='http://www.w3.org/1998/Math/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 <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>∞Grpd</a>: the <a class='existingWikiWord' href='/nlab/show/diff/direct+image'>direct image</a> is the <a class='existingWikiWord' href='/nlab/show/diff/global+section'>global section</a>s <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functor</a> <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Γ</mi></mrow><annotation encoding='application/x-tex'>\Gamma</annotation></semantics></math>, the <a class='existingWikiWord' href='/nlab/show/diff/inverse+image'>inverse image</a> is the <a class='existingWikiWord' href='/nlab/show/diff/constant+infinity-stack'>constant ∞-stack</a> functor</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>LConst</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mstyle mathvariant='bold'><mi>H</mi></mstyle><munderover><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>⊥</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow><munder><mo>⟶</mo><mi>Γ</mi></munder><mover><mo>⟵</mo><mi>LConst</mi></mover></munderover><mn>∞</mn><mi>Grpd</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> (LConst \dashv \Gamma) \;\colon\; \mathbf{H} \underoverset {\underset{\Gamma}{\longrightarrow}} {\overset{LConst}{\longleftarrow}} {\;\;\; \bot \;\;\;} \infty Grpd \,. </annotation></semantics></math></div> <p>In fact, this is unique, up to <a class='existingWikiWord' href='/nlab/show/diff/equivalence+in+an+%28infinity%2C1%29-category'>equivalence</a>: Since every <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoid is an <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>$(\infty,1)$-colimit</a> (namely over itself, by <a href='infinity1-limit#EveryInfinityGroupoidIsHomotopyColimitOfConstantFunctorOverItself'>this Prop.</a>) of the <a class='existingWikiWord' href='/nlab/show/diff/point'>point</a> (hence of the <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a>), and since the <a class='existingWikiWord' href='/nlab/show/diff/inverse+image'>inverse image</a> <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-functor <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>LConst</mi></mrow><annotation encoding='application/x-tex'>LConst</annotation></semantics></math> needs to preserve these <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-colimits (being a <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a>) as well as the point (being a <a class='existingWikiWord' href='/nlab/show/diff/exact+functor'>lex functor</a>).</p> <h3 id='ClosedMonoidalStructure'>Closed monoidal structure</h3> <div class='num_prop'> <h6 id='proposition_3'>Proposition</h6> <p>Every <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_47' xmlns='http://www.w3.org/1998/Math/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 is a <a class='existingWikiWord' href='/nlab/show/diff/cartesian+closed+%28infinity%2C1%29-category'>cartesian closed (∞,1)-category</a>.</p> </div> <div class='proof'> <h6 id='proof'>Proof</h6> <p>By the fact that every <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_48' xmlns='http://www.w3.org/1998/Math/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 <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> has <a class='existingWikiWord' href='/nlab/show/diff/pullback-stable+colimit'>universal colimits</a> it follows that for every object <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functor</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>:</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo>→</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'> X \times (-) : \mathbf{H} \to \mathbf{H} </annotation></semantics></math></div> <p>preserves all <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>(∞,1)-colimit</a>s. Since every <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_52' xmlns='http://www.w3.org/1998/Math/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 is a <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>locally presentable (∞,1)-category</a> it follows with the <a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor+theorem'>adjoint (∞,1)-functor theorem</a> that there is a <a class='existingWikiWord' href='/nlab/show/diff/right+adjoint'>right</a> <a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>adjoint (∞,1)-functor</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>⊣</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mo>:</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mover><munder><mo>→</mo><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo></mrow></munder><mover><mo>←</mo><mrow><mi>X</mi><mo>×</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow></mover></mover><mstyle mathvariant='bold'><mi>H</mi></mstyle><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> (X \times (-) \dashv [X,-]) : \mathbf{H} \stackrel{\overset{X \times (-)}{\leftarrow}}{\underset{[X,-]}{\to}} \mathbf{H} \,. </annotation></semantics></math></div></div> <div class='num_prop'> <h6 id='proposition_4'>Proposition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-site'>(∞,1)-site</a> for <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> we have that the <a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom</a> (<a class='existingWikiWord' href='/nlab/show/diff/mapping+stack'>mapping stack</a>) <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[X,-]</annotation></semantics></math> is given on <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∈</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>A \in \mathbf{H}</annotation></semantics></math> by the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-sheaf'>(∞,1)-sheaf</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo><mspace width='thinmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thinmathspace' /><mi>U</mi><mo>↦</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mi>L</mi><mi>y</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> [X,A] \,\colon\, U \mapsto \mathbf{H}(X \times L y(U), A) \,, </annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>:</mo><mi>C</mi><mo>→</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>y : C \to \mathbf{H}</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories'>(∞,1)-Yoneda embedding</a> and <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mo>:</mo><msub><mi>PSh</mi> <mi>C</mi></msub><mo>→</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>L : PSh_C \to \mathbf{H}</annotation></semantics></math> denotes <a class='existingWikiWord' href='/nlab/show/diff/infinity-stackification'>∞-stackification</a>.</p> </div> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>The argument is entirely analogous to that of the <a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+structure+on+sheaves'>closed monoidal structure on sheaves</a>.</p> <p>We use the <a class='existingWikiWord' href='/nlab/show/diff/fully+faithful+%28infinity%2C1%29-functor'>full and faithful</a> <a class='existingWikiWord' href='/nlab/show/diff/geometric+embedding'>geometric embedding</a> <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>L</mi><mo>⊣</mo><mi>i</mi><mo stretchy='false'>)</mo><mo>:</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo>↪</mo><msub><mi>PSh</mi> <mi>C</mi></msub></mrow><annotation encoding='application/x-tex'>(L \dashv i) : \mathbf{H} \hookrightarrow PSh_C</annotation></semantics></math> and the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories'>(∞,1)-Yoneda lemma</a> to find for all <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>U \in C</annotation></semantics></math> the value</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>≃</mo><msub><mi>PSh</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>y</mi><mi>U</mi><mo>,</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> [X,A](U) \simeq PSh_C(y U, [X,A]) </annotation></semantics></math></div> <p>and then the fact that <a class='existingWikiWord' href='/nlab/show/diff/infinity-stackification'>∞-stackification</a> <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a> to inclusion to get</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi><mo>≃</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mi>L</mi><mi>y</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>,</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \cdots \simeq \mathbf{H}(L y(U), [X,A]) \,. </annotation></semantics></math></div> <p>Then the defining adjunction <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>⊣</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X \times (-) \dashv [X,-])</annotation></semantics></math> gives</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi><mo>≃</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mi>L</mi><mi>y</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \cdots \simeq \mathbf{H}(X \times L y(U) , A) \,. </annotation></semantics></math></div></div> <h3 id='PoweringOfInfinityToposesOverInfinityGroupoids'>Powering of <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-toposes over <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoids</h3> <p>We discuss how the <a class='existingWikiWord' href='/nlab/show/diff/powering'>powering</a> of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>$\infty$-toposes</a> over <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>$Grpd_\infty$</a> is given by forming <a class='existingWikiWord' href='/nlab/show/diff/mapping+stack'>mapping stacks</a> out of <a class='existingWikiWord' href='/nlab/show/diff/locally+constant+infinity-stack'>locally constant $\infty$-stacks</a>. All of the following formulas and their proofs hold verbatim also for <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+topos'>Grothendieck toposes</a>, as they just use general abstract properties.</p> <p>\linebreak</p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> be an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>$\infty$-topos</a></p> <ul> <li> <p>with terminal <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-geometric+morphism'>geometric morphism</a> denoted</p> <div class='maruku-equation' id='eq:TerminalGeometricMorphismAdjunction'><span class='maruku-eq-number'>(1)</span><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle><munderover><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>⊥</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow><munder><mo>⟶</mo><mi>Γ</mi></munder><mover><mo>⟵</mo><mi>LConst</mi></mover></munderover><msub><mi>Grp</mi> <mn>∞</mn></msub><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> \mathbf{H} \underoverset {\underset{\Gamma}{\longrightarrow}} {\overset{LConst}{\longleftarrow}} {\;\;\;\;\bot\;\;\;\;} Grp_\infty \,, </annotation></semantics></math></div> <p>where the <a class='existingWikiWord' href='/nlab/show/diff/inverse+image'>inverse image</a> constructs <a class='existingWikiWord' href='/nlab/show/diff/locally+constant+infinity-stack'>locally constant $\infty$-stacks</a>,</p> </li> <li> <p>and with its <a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom</a> (<a class='existingWikiWord' href='/nlab/show/diff/mapping+stack'>mapping stack</a>) <a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>adjunction</a> denoted</p> <div class='maruku-equation' id='eq:MappingStackAdjunction'><span class='maruku-eq-number'>(2)</span><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle><munderover><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>⊥</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow><munder><mo>⟶</mo><mrow><mi>Maps</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow></munder><mover><mo>⟵</mo><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>×</mo><mi>X</mi></mrow></mover></munderover><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'> \mathbf{H} \underoverset {\underset{Maps(X,-)}{\longrightarrow}} { \overset{ (-) \times X }{\longleftarrow} } {\;\;\;\; \bot \;\;\;\;} \mathbf{H} </annotation></semantics></math></div> <p>for <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mspace width='thinmathspace' /><mo>∈</mo><mspace width='thinmathspace' /><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>X \,\in\, \mathbf{H}</annotation></semantics></math>.</p> <p>Notice that this construction is also <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>$\infty$-functorial</a> in the first argument: <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Maps</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi><mo>,</mo><mspace width='thinmathspace' /><mi>A</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'>Maps\big( X \xrightarrow{f} Y ,\, A \big)</annotation></semantics></math> is the morphism which under the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories'>$\infty$-Yoneda lemma</a> over <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> (which is large but locally small, so that the lemma does apply) corresponds to</p> </li> </ul> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace' /><mi>Maps</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>×</mo><mi>X</mi><mo>,</mo><mspace width='thinmathspace' /><mi>A</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mover><mo>→</mo><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>×</mo><mi>f</mi><mo>,</mo><mspace width='thinmathspace' /><mi>A</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow></mover><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>×</mo><mi>Y</mi><mo>,</mo><mspace width='thinmathspace' /><mi>A</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace' /><mi>Maps</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \mathbf{H} \big( (-) ,\, Maps(X,A) \big) \;\simeq\; \mathbf{H} \big( (-) \times X ,\, A \big) \xrightarrow{ \mathbf{H} \big( (-) \times f ,\, A \big) } \mathbf{H} \big( (-) \times Y ,\, A \big) \;\simeq\; \mathbf{H} \big( (-) ,\, Maps(X,A) \big) \,. </annotation></semantics></math></div> <p>By definition, for any <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>∈</mo><msub><mi>Grpd</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>S \in Grpd_\infty</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>X \in \mathbf{H}</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/powering'>powering</a>] is the <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>(∞,1)-limit</a> over the <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a> constant on <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mi>K</mi></msup><mspace width='thinmathspace' /><mo>=</mo><mspace width='thinmathspace' /><msub><mrow><munder><mi>lim</mi> <mo>←</mo></munder></mrow> <mi>K</mi></msub><mi>X</mi></mrow><annotation encoding='application/x-tex'> X^K \,=\, {\lim_\leftarrow}_K X </annotation></semantics></math></div> <p>while the <a class='existingWikiWord' href='/nlab/show/diff/copower'>tensoring</a> is the <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>(∞,1)-colimit</a> over the diagram constant on <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>⋅</mo><mi>X</mi><mspace width='thinmathspace' /><mo>=</mo><mspace width='thinmathspace' /><msub><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow> <mi>K</mi></msub><mi>X</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> K \cdot X \,=\, {\lim_{\to}}_K X \,. </annotation></semantics></math></div> <p>\begin{remark} Under <a class='existingWikiWord' href='/nlab/show/diff/Isbell+duality'>Isbell duality</a>, the powering operations on homotopy types <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> corresponds to higher order <a class='existingWikiWord' href='/nlab/show/diff/Hochschild+cohomology'>Hochschild cohomology</a> of suitable algebras of functions on <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, as discussed there. \end{remark}</p> <p>\begin{proposition} The <em>powering</em> of <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> over <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>$Grpd_\infty$</a> is given by the <a class='existingWikiWord' href='/nlab/show/diff/mapping+stack'>mapping stack</a> out of the <a class='existingWikiWord' href='/nlab/show/diff/locally+constant+infinity-stack'>locally constant $\infty$-stacks</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msubsup><mi>Grpd</mi> <mn>∞</mn> <mi>op</mi></msubsup><mo>×</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mtd> <mtd><mover><mo>⟶</mo><mrow><msup><mi>LConst</mi> <mi>op</mi></msup><mo>×</mo><mi mathvariant='normal'>id</mi></mrow></mover></mtd> <mtd><msup><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mi>op</mi></msup><mo>×</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>Maps</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow></mover></mtd> <mtd><mstyle mathvariant='bold'><mi>H</mi></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ Grpd_\infty^{op} \times \mathbf{H} & \overset{ LConst^{op} \times \mathrm{id} }{\longrightarrow} & \mathbf{H}^{op} \times \mathbf{H} & \overset{Maps(-,-)}{\longrightarrow} & \mathbf{H} } </annotation></semantics></math></div> <p>in that this operation has the following properties:</p> <ol> <li> <p>For all <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>,</mo><mspace width='thinmathspace' /><mi>A</mi><mspace width='thinmathspace' /><mo>∈</mo><mspace width='thinmathspace' /><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>X,\,A \,\in\, \mathbf{H}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mspace width='thinmathspace' /><mo>∈</mo><mspace width='thinmathspace' /><msub><mi>Grpd</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>S \,\in\, Grpd_\infty</annotation></semantics></math> we have a <a class='existingWikiWord' href='/nlab/show/diff/natural+equivalence'>natural equivalence</a></p> <div class='maruku-equation' id='eq:eq3'><span class='maruku-eq-number'>(3)</span><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='1.8em' minsize='1.8em'>(</mo><mi>X</mi><mo>,</mo><mspace width='thinmathspace' /><mi>Maps</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>LConst</mi><mo stretchy='false'>(</mo><mi>S</mi><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace' /><mi>A</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><msub><mi>Grpd</mi> <mn>∞</mn></msub><mo maxsize='1.8em' minsize='1.8em'>(</mo><mi>S</mi><mo>,</mo><mspace width='thinmathspace' /><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>X</mi><mo>,</mo><mspace width='thinmathspace' /><mi>A</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo></mrow><annotation encoding='application/x-tex'> \mathbf{H} \Big( X ,\, Maps \big( LConst(S) ,\, A \big) \Big) \;\; \simeq \;\; Grpd_\infty \Big( S ,\, \mathbf{H} \big( X ,\, A \big) \Big) </annotation></semantics></math></div></li> <li> <p>In its first argument the operation</p> <ol> <li> <p>sends the <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a> (the <a class='existingWikiWord' href='/nlab/show/diff/point'>point</a>) to the identity:</p> <div class='maruku-equation' id='eq:MappingStackOutOfLocallyConstantPreservesLimitsInFirstArg'><span class='maruku-eq-number'>(4)</span><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Maps</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>LConst</mi><mo stretchy='false'>(</mo><mo>*</mo><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace' /><mi>X</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>X</mi></mrow><annotation encoding='application/x-tex'> Maps \big( LConst(\ast) ,\, X \big) \;\; \simeq \;\; X </annotation></semantics></math></div></li> <li> <p>sends <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>$\infty$-colimits</a> to <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>$\infty$-limits</a>:</p> <div class='maruku-equation' id='eq:MappingStackOutOfLConstStacksPreservesColimitInFirstArgument'><span class='maruku-eq-number'>(5)</span><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Maps</mi><mo maxsize='1.8em' minsize='1.8em'>(</mo><munder><mi>lim</mi><mo>⟶</mo></munder><mspace width='thinmathspace' /><mi>LConst</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo>,</mo><mspace width='thinmathspace' /><mi>X</mi><mo maxsize='1.8em' minsize='1.8em'>)</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><munder><mi>lim</mi><mo>⟵</mo></munder><mspace width='thinmathspace' /><mi>Maps</mi><mo maxsize='1.8em' minsize='1.8em'>(</mo><mi>LConst</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo>,</mo><mspace width='thinmathspace' /><mi>X</mi><mo maxsize='1.8em' minsize='1.8em'>)</mo><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> Maps \Big( \underset{ \longrightarrow }{\lim} \, LConst\big(S_\bullet\big) ,\, X \Big) \;\; \simeq \;\; \underset{ \longleftarrow }{\lim} \, Maps \Big( LConst\big(S_\bullet\big) ,\, X \Big) \,, </annotation></semantics></math></div></li> </ol> <p>where all <a class='existingWikiWord' href='/nlab/show/diff/equivalence+in+an+%28infinity%2C1%29-category'>equivalences</a> shown are <a class='existingWikiWord' href='/nlab/show/diff/natural+equivalence'>natural</a>.</p> </li> </ol> <p>\end{proposition}</p> <p>\begin{proof}</p> <p>For the first statement to be proven, consider the following sequence of <a class='existingWikiWord' href='/nlab/show/diff/natural+equivalence'>natural equivalences</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mtable columnalign='left left left' displaystyle='false' rowspacing='0.5ex'><mtr><mtd><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='1.8em' minsize='1.8em'>(</mo><mi>X</mi><mo>,</mo><mspace width='thinmathspace' /><mi>Maps</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>LConst</mi><mo stretchy='false'>(</mo><mi>S</mi><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace' /><mi>A</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo></mtd> <mtd><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>X</mi><mo>×</mo><mi>LConst</mi><mo stretchy='false'>(</mo><mi>S</mi><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace' /><mi>A</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd> <mtd><mtext>(eq:MappingStackAdjunction)</mtext></mtd></mtr> <mtr><mtd /> <mtd><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='1.8em' minsize='1.8em'>(</mo><mi>LConst</mi><mo stretchy='false'>(</mo><mi>S</mi><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace' /><mi>Maps</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>X</mi><mo>,</mo><mspace width='thinmathspace' /><mi>A</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo></mtd> <mtd><mtext>(eq:MappingStackAdjunction)</mtext></mtd></mtr> <mtr><mtd /> <mtd><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><msub><mi>Grpd</mi> <mn>∞</mn></msub><mo maxsize='1.8em' minsize='1.8em'>(</mo><mi>S</mi><mo>,</mo><mspace width='thinmathspace' /><mi>Γ</mi><mspace width='thinmathspace' /><mi>Maps</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>X</mi><mo>,</mo><mspace width='thinmathspace' /><mi>A</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo></mtd> <mtd><mtext> (eq:TerminalGeometricMorphismAdjunction) </mtext></mtd></mtr> <mtr><mtd /> <mtd><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><msub><mi>Grpd</mi> <mn>∞</mn></msub><mo maxsize='2.4em' minsize='2.4em'>(</mo><mi>S</mi><mo>,</mo><mspace width='thinmathspace' /><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='1.8em' minsize='1.8em'>(</mo><msub><mo>*</mo> <mstyle mathvariant='bold'><mi>H</mi></mstyle></msub><mo>,</mo><mspace width='thinmathspace' /><mi>Maps</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>X</mi><mo>,</mo><mspace width='thinmathspace' /><mi>A</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo><mo maxsize='2.4em' minsize='2.4em'>)</mo></mtd> <mtd><mtext>by</mtext><mspace width='thickmathspace' /><mtext><a href='https://ncatlab.org/nlab/show/terminal+geometric+morphism#DirectImageOfTerminalGeometricMoprhismIsHomOutOfTerminalObject'>this Prop.</a></mtext></mtd></mtr> <mtr><mtd /> <mtd><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><msub><mi>Grpd</mi> <mn>∞</mn></msub><mo maxsize='1.8em' minsize='1.8em'>(</mo><mi>S</mi><mo>,</mo><mspace width='thinmathspace' /><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='1.2em' minsize='1.2em'>(</mo><msub><mo>*</mo> <mstyle mathvariant='bold'><mi>H</mi></mstyle></msub><mo>×</mo><mi>X</mi><mo>,</mo><mspace width='thinmathspace' /><mi>A</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo></mtd> <mtd><mtext>(eq:MappingStackAdjunction)</mtext></mtd></mtr> <mtr><mtd /> <mtd><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><msub><mi>Grpd</mi> <mn>∞</mn></msub><mo maxsize='1.8em' minsize='1.8em'>(</mo><mi>S</mi><mo>,</mo><mspace width='thinmathspace' /><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>X</mi><mo>,</mo><mspace width='thinmathspace' /><mi>A</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo></mtd></mtr></mtable></mrow><annotation encoding='application/x-tex'> \begin{array}{lll} \mathbf{H} \Big( X ,\, Maps \big( LConst(S) ,\, A \big) \Big) & \;\simeq\; \mathbf{H} \big( X \times LConst(S) ,\, A \big) & \text{(eq:MappingStackAdjunction)} \\ & \;\simeq\; \mathbf{H} \Big( LConst(S) ,\, Maps \big( X ,\, A \big) \Big) & \text{(eq:MappingStackAdjunction)} \\ & \;\simeq\; Grpd_\infty \Big( S ,\, \Gamma \, Maps \big( X ,\, A \big) \Big) & \text{ (eq:TerminalGeometricMorphismAdjunction) } \\ & \;\simeq\; Grpd_\infty \bigg( S ,\, \mathbf{H} \Big( \ast_{\mathbf{H}} ,\, Maps \big( X ,\, A \big) \Big) \bigg) & \text{by}\;\text{<a href="https://ncatlab.org/nlab/show/terminal+geometric+morphism#DirectImageOfTerminalGeometricMoprhismIsHomOutOfTerminalObject">this Prop.</a>} \\ & \;\simeq\; Grpd_\infty \Big( S ,\, \mathbf{H} \big( \ast_{\mathbf{H}} \times X ,\, A \big) \Big) & \text{(eq:MappingStackAdjunction)} \\ & \;\simeq\; Grpd_\infty \Big( S ,\, \mathbf{H} \big( X ,\, A \big) \Big) \end{array} </annotation></semantics></math></div> <p>For the second statement, recall that <a class='existingWikiWord' href='/nlab/show/diff/hom-functor+preserves+limits'>hom-functors preserve limits</a> in that there are <a class='existingWikiWord' href='/nlab/show/diff/natural+equivalence'>natural</a> <a class='existingWikiWord' href='/nlab/show/diff/equivalence+in+an+%28infinity%2C1%29-category'>equivalences</a> of the form</p> <div class='maruku-equation' id='eq:HomFunctorPreservesLimits'><span class='maruku-eq-number'>(6)</span><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='1.8em' minsize='1.8em'>(</mo><munder><mi>lim</mi><munder><mo>⟶</mo><mi>i</mi></munder></munder><mspace width='thinmathspace' /><mo>,</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><mspace width='thinmathspace' /><munder><mi>lim</mi><munder><mo>⟵</mo><mi>j</mi></munder></munder><mspace width='thinmathspace' /><mo>,</mo><msub><mi>A</mi> <mi>j</mi></msub><mo maxsize='1.8em' minsize='1.8em'>)</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><munder><mi>lim</mi><munder><mo>⟵</mo><mi>i</mi></munder></munder><mspace width='thinmathspace' /><munder><mi>lim</mi><munder><mo>⟵</mo><mi>j</mi></munder></munder><mspace width='thinmathspace' /><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='1.8em' minsize='1.8em'>(</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><mspace width='thinmathspace' /><msub><mi>A</mi> <mi>j</mi></msub><mo maxsize='1.8em' minsize='1.8em'>)</mo><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> \mathbf{H} \Big( \underset{\underset{i}{\longrightarrow}}{\lim} \,, X_i ,\, \underset{\underset{j}{\longleftarrow}}{\lim} \,, A_j \Big) \;\; \simeq \;\; \underset{\underset{i}{\longleftarrow}}{\lim} \, \underset{\underset{j}{\longleftarrow}}{\lim} \, \mathbf{H} \Big( X_i ,\, A_j \Big) \,, </annotation></semantics></math></div> <p>and that <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-toposes have <a class='existingWikiWord' href='/nlab/show/diff/pullback-stable+colimit'>universal colimits</a>, in particular that the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a> operation is a <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a> <a class='maruku-eqref' href='#eq:MappingStackAdjunction'>(2)</a> and <a class='existingWikiWord' href='/nlab/show/diff/adjoints+preserve+%28co-%29limits'>hence preserves colimits</a>:</p> <div class='maruku-equation' id='eq:ProductsPreserveColimits'><span class='maruku-eq-number'>(7)</span><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>×</mo><mspace width='thinmathspace' /><munder><mi>lim</mi><mo>⟶</mo></munder><mspace width='thinmathspace' /><msub><mi>S</mi> <mo>•</mo></msub><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><munder><mi>lim</mi><mo>⟶</mo></munder><mspace width='thinmathspace' /><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>×</mo><mspace width='thinmathspace' /><msub><mi>S</mi> <mo>•</mo></msub><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> (-) \,\times\, \underset{{\longrightarrow}}{\lim} \, S_\bullet \;\; \simeq \;\; \underset{{\longrightarrow}}{\lim} \, \big( (-) \,\times\, S_\bullet \big) \,. </annotation></semantics></math></div> <p>With this, we get the following sequences of <a class='existingWikiWord' href='/nlab/show/diff/natural+equivalence'>natural equivalences</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mtable columnalign='left left left' displaystyle='false' rowspacing='0.5ex'><mtr><mtd /> <mtd><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='2.4em' minsize='2.4em'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace' /><mi>Maps</mi><mo maxsize='1.8em' minsize='1.8em'>(</mo><munder><mi>lim</mi><mo>⟶</mo></munder><mspace width='thinmathspace' /><mi>LConst</mi><mo stretchy='false'>(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace' /><mi>X</mi><mo maxsize='1.8em' minsize='1.8em'>)</mo><mo maxsize='2.4em' minsize='2.4em'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='1.8em' minsize='1.8em'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>×</mo><munder><mi>lim</mi><mo>⟶</mo></munder><mspace width='thinmathspace' /><mi>LConst</mi><mo stretchy='false'>(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace' /><mi>X</mi><mo maxsize='1.8em' minsize='1.8em'>)</mo></mtd> <mtd><mtext> (eq:MappingStackAdjunction) </mtext></mtd></mtr> <mtr><mtd /> <mtd><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='1.8em' minsize='1.8em'>(</mo><munder><mi>lim</mi><mo>⟶</mo></munder><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>×</mo><mi>LConst</mi><mo stretchy='false'>(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo>,</mo><mspace width='thinmathspace' /><mi>X</mi><mo maxsize='1.8em' minsize='1.8em'>)</mo></mtd> <mtd><mtext> (eq:ProductsPreserveColimits) </mtext></mtd></mtr> <mtr><mtd /> <mtd><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><munder><mi>lim</mi><mo>⟵</mo></munder><mspace width='thinmathspace' /><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>×</mo><mi>LConst</mi><mo stretchy='false'>(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace' /><mi>X</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd> <mtd><mtext> (eq:HomFunctorPreservesLimits) </mtext></mtd></mtr> <mtr><mtd /> <mtd><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><munder><mi>lim</mi><mo>⟵</mo></munder><mspace width='thinmathspace' /><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='1.8em' minsize='1.8em'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace' /><mi>Maps</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>LConst</mi><mo stretchy='false'>(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace' /><mi>X</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo></mtd> <mtd><mtext> (eq:MappingStackAdjunction) </mtext></mtd></mtr> <mtr><mtd /> <mtd><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo maxsize='1.8em' minsize='1.8em'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace' /><munder><mi>lim</mi><mo>⟵</mo></munder><mspace width='thinmathspace' /><mi>Maps</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>LConst</mi><mo stretchy='false'>(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace' /><mi>X</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo></mtd> <mtd><mtext> (eq:HomFunctorPreservesLimits) </mtext><mspace width='thinmathspace' /><mo>.</mo></mtd></mtr></mtable></mrow><annotation encoding='application/x-tex'> \begin{array}{lll} & \mathbf{H} \bigg( (-) ,\, Maps \Big( \underset{\longrightarrow}{\lim} \, LConst(S_\bullet) ,\, X \Big) \bigg) \\ & \;\simeq\; \mathbf{H} \Big( (-) \times \underset{\longrightarrow}{\lim} \, LConst(S_\bullet) ,\, X \Big) & \text{ (eq:MappingStackAdjunction) } \\ & \;\simeq\; \mathbf{H} \Big( \underset{\longrightarrow}{\lim} \big( (-) \times LConst(S_\bullet) \big) ,\, X \Big) & \text{ (eq:ProductsPreserveColimits) } \\ & \;\simeq\; \underset{\longleftarrow}{\lim} \, \mathbf{H} \big( (-) \times LConst(S_\bullet) ,\, X \big) & \text{ (eq:HomFunctorPreservesLimits) } \\ & \;\simeq\; \underset{\longleftarrow}{\lim} \, \mathbf{H} \Big( (-) ,\, Maps \big( LConst(S_\bullet) ,\, X \big) \Big) & \text{ (eq:MappingStackAdjunction) } \\ & \;\simeq\; \mathbf{H} \Big( (-) ,\, \underset{\longleftarrow}{\lim} \, Maps \big( LConst(S_\bullet) ,\, X \big) \Big) & \text{ (eq:HomFunctorPreservesLimits) } \,. \end{array} </annotation></semantics></math></div> <p>This implies <a class='maruku-eqref' href='#eq:MappingStackOutOfLConstStacksPreservesColimitInFirstArgument'>(5)</a> by the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories'>$\infty$-Yoneda lemma</a> over <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> (which is large but locally small, so that the lemma does apply).</p> <p>Finally <a class='maruku-eqref' href='#eq:MappingStackOutOfLocallyConstantPreservesLimitsInFirstArg'>(4)</a> is immediate from the fact that <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>LConst</mi></mrow><annotation encoding='application/x-tex'>LConst</annotation></semantics></math> preserves the terminal object, by definition:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Maps</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>LConst</mi><mo stretchy='false'>(</mo><mo>*</mo><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thinmathspace' /><mi>X</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mi>Maps</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><msub><mo>*</mo> <mstyle mathvariant='bold'><mi>H</mi></mstyle></msub><mo>,</mo><mspace width='thinmathspace' /><mi>X</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mi>X</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> Maps \big( LConst(\ast) ,\, X \big) \;\simeq\; Maps \big( \ast_{\mathbf{H}} ,\, X \big) \;\simeq\; X \,. </annotation></semantics></math></div> <p>\end{proof}</p> <h3 id='slicetoposes'>Slice-<math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_100' xmlns='http://www.w3.org/1998/Math/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>-toposes</h3> <div class='num_prop'> <h6 id='proposition_5'>Proposition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> an <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_102' xmlns='http://www.w3.org/1998/Math/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 and <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>X \in \mathbf{H}</annotation></semantics></math> an object, the <a class='existingWikiWord' href='/nlab/show/diff/over-%28infinity%2C1%29-category'>slice-(∞,1)-category</a> <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mrow><mo stretchy='false'>/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\mathbf{H}_{/X}</annotation></semantics></math> is itself an <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_105' xmlns='http://www.w3.org/1998/Math/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 – an <strong><a class='existingWikiWord' href='/nlab/show/diff/over-%28infinity%2C1%29-topos'>over-(∞,1)-topos</a></strong>. The projection <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mo>!</mo></msub><mo>:</mo><msub><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mrow><mo stretchy='false'>/</mo><mi>X</mi></mrow></msub><mo>→</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\pi_! : \mathbf{H}_{/X} \to \mathbf{H}</annotation></semantics></math> part of an <a class='existingWikiWord' href='/nlab/show/diff/essential+geometric+morphism'>essential geometric morphism</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>π</mi><mo>:</mo><msub><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mrow><mo stretchy='false'>/</mo><mi>X</mi></mrow></msub><mover><mover><munder><mo>→</mo><mrow><msub><mi>π</mi> <mo>*</mo></msub></mrow></munder><mover><mo>←</mo><mrow><msup><mi>π</mi> <mo>*</mo></msup></mrow></mover></mover><mover><mo>→</mo><mrow><msub><mi>π</mi> <mo>!</mo></msub></mrow></mover></mover><mstyle mathvariant='bold'><mi>H</mi></mstyle><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \pi : \mathbf{H}_{/X} \stackrel{\overset{\pi_!}{\to}}{\stackrel{\overset{\pi^*}{\leftarrow}}{\underset{\pi_*}{\to}}} \mathbf{H} \,. </annotation></semantics></math></div></div> <p>This is <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>HTT, prop. 6.3.5.1</a>.</p> <p>The <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_108' xmlns='http://www.w3.org/1998/Math/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 <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mrow><mo stretchy='false'>/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\mathbf{H}_{/X}</annotation></semantics></math> could be called the <a class='existingWikiWord' href='/nlab/show/diff/big+and+little+toposes'>gros topos</a> of <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. A <a class='existingWikiWord' href='/nlab/show/diff/geometric+morphism'>geometric morphism</a> <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>K</mi></mstyle><mo>→</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{K} \to \mathbf{H}</annotation></semantics></math> that factors as <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>K</mi></mstyle><mover><mo>→</mo><mo>≃</mo></mover><msub><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mrow><mo stretchy='false'>/</mo><mi>X</mi></mrow></msub><mover><mo>→</mo><mi>π</mi></mover><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{K} \xrightarrow{\simeq} \mathbf{H}_{/X} \stackrel{\pi}{\to} \mathbf{H}</annotation></semantics></math> is called an <a class='existingWikiWord' href='/nlab/show/diff/%C3%A9tale+geometric+morphism'>etale geometric morphism</a>.</p> <h3 id='syntax_in_univalent_homotopy_type_theory'>Syntax in univalent homotopy type theory</h3> <p><math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_113' xmlns='http://www.w3.org/1998/Math/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>-Toposes provide <a class='existingWikiWord' href='/nlab/show/diff/categorical+semantics'>categorical semantics</a> for <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a> with a <a class='existingWikiWord' href='/nlab/show/diff/univalence+axiom'>univalent</a> Tarskian <a class='existingWikiWord' href='/nlab/show/diff/type+universe'>type of types</a> (which inteprets as the <a class='existingWikiWord' href='/nlab/show/diff/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos'>object classifier</a>).</p> <p>For more on this see at</p> <ul> <li> <p><em><a class='existingWikiWord' href='/homotopytypetheory/show/diff/model+of+type+theory+in+an+%28infinity%2C1%29-topos+%3E+history' title='homotopytypetheory'>model of type theory in an (infinity,1)-topos</a></em></p> </li> <li> <p><em><a href='relation+between+type+theory+and+category+theory#HomotopyWithUnivalence'>relation between type theory and category theory – Univalent homotopy type theory and infinity-toposes</a></em></p> </li> </ul> <h2 id='ToposTheory'><math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_114' xmlns='http://www.w3.org/1998/Math/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</h2> <p>Most of the standard constructions in <a class='existingWikiWord' href='/nlab/show/diff/sheaf+and+topos+theory'>topos theory</a> have or should have immediate generalizations to the context of <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_115' xmlns='http://www.w3.org/1998/Math/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>-toposes, since all notions of <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a> exist for <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-categories</a>.</p> <p>For instance there are evident notions of</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/geometric+morphism'>geometric morphism</a>s between <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_116' xmlns='http://www.w3.org/1998/Math/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>-toposes, such as the <a class='existingWikiWord' href='/nlab/show/diff/global+section'>global section</a> geometric morphism to the <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal</a> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-sheaf</a> <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_117' xmlns='http://www.w3.org/1998/Math/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 <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>∞Grpd</a>.</li> </ul> <p>Moreover, it turns out that <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_118' xmlns='http://www.w3.org/1998/Math/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>-toposes come with plenty of internal structures, more than canonically present in an ordinary topos. Every <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_119' xmlns='http://www.w3.org/1998/Math/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 comes with its intrinsic notion of</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology in an (∞,1)-topos</a></li> </ul> <p>and with an intrinsic notion of</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/homotopy+groups+in+an+%28infinity%2C1%29-topos'>homotopy in an (∞,1)-topos</a>.</li> </ul> <p>In classical topos theory, cohomology and homotopy of a topos <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> are defined in terms of <a class='existingWikiWord' href='/nlab/show/diff/simplicial+object'>simplicial object</a>s in <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>. If <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+topos'>sheaf topos</a> with <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a> <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> and <a class='existingWikiWord' href='/nlab/show/diff/point+of+a+topos'>enough point</a>s, then this classical construction is secretly really a model for the intrinsic cohomology and homotopy in the above sense of the <a class='existingWikiWord' href='/nlab/show/diff/hypercomplete+%28infinity%2C1%29-topos'>hypercomplete (∞,1)-topos</a> of <a class='existingWikiWord' href='/nlab/show/diff/infinity-stack'>∞-stack</a>s on <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>.</p> <p>The beginning of a list of all the structures that exist intrinsically in a big <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_125' xmlns='http://www.w3.org/1998/Math/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 is at</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/cohesive+%28infinity%2C1%29-topos'>cohesive (∞,1)-topos</a>.</li> </ul> <p>But <strong><math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_126' xmlns='http://www.w3.org/1998/Math/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</strong> in the style of an <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-analog of the <a class='existingWikiWord' href='/nlab/show/diff/Sketches+of+an+Elephant'>Elephant</a> is only barely beginning to be conceived.</p> <p>There are some indications as to what the</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/internal+logic+of+an+%28infinity%2C1%29-topos'>internal logic of an (∞,1)-topos</a></li> </ul> <p>should be.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/elementary+%28infinity%2C1%29-topos'>elementary (∞,1)-topos</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-pretopos'>(∞,1)-pretopos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+topos'>model topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/structured+%28infinity%2C1%29-topos'>structured (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/proper+geometric+morphism'>compact topos</a>, <a class='existingWikiWord' href='/nlab/show/diff/coherent+%28infinity%2C1%29-topos'>coherent (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category+object+in+an+%28infinity%2C1%29-category'>category object in an (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tangent+%28infinity%2C1%29-category'>tangent (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+topos'>doubly monoidal (∞,1)-topos</a></p> </li> </ul> <p><strong>flavors of <a class='existingWikiWord' href='/nlab/show/diff/higher+topos+theory'>higher toposes</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/1-topos'>1-topos</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%280%2C1%29-topos'>(0,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/1-topos'>(1,1)-topos</a> = <a class='existingWikiWord' href='/nlab/show/diff/topos'>topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-topos'>(2,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28n%2C1%29-topos'>$(n,1)$-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a> (<a class='existingWikiWord' href='/nlab/show/diff/n-localic+%28infinity%2C1%29-topos'>$n$-localic</a>)</p> <p><math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow><annotation encoding='application/x-tex'>\;\;\;\;</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/model+topos'>model topos</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-topos'>2-topos</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-topos'>(2,2)-topos</a> (<a class='existingWikiWord' href='/nlab/show/diff/n-localic+2-topos'>$n$-localic</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C2%29-topos'>(∞,2)-topos</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/n-topos'>$n$-topos</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2Cn%29-topos'>$(\infty,n)$-topos</a></li> </ul> </li> </ul> <p><strong>Locally presentable categories:</strong> <a class='existingWikiWord' href='/nlab/show/diff/cocomplete+category'>Cocomplete</a> possibly-<a class='existingWikiWord' href='/nlab/show/diff/large+category'>large categories</a> generated under <a class='existingWikiWord' href='/nlab/show/diff/filtered+colimit'>filtered colimits</a> by <a class='existingWikiWord' href='/nlab/show/diff/small+object'>small</a> <a class='existingWikiWord' href='/nlab/show/diff/generator'>generators</a> under <a class='existingWikiWord' href='/nlab/show/diff/small+limit'>small</a> <a class='existingWikiWord' href='/nlab/show/diff/relation'>relations</a>. Equivalently, <a class='existingWikiWord' href='/nlab/show/diff/accessible+functor'>accessible</a> <a class='existingWikiWord' href='/nlab/show/diff/reflective+localization'>reflective localizations</a> of <a class='existingWikiWord' href='/nlab/show/diff/free+cocompletion'>free cocompletions</a>. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a <a class='existingWikiWord' href='/nlab/show/diff/exact+functor'>left exact</a> localization.</p> <table><thead><tr><th><math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/%28n%2Cr%29-category'>(n,r)-categories</a><math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></th><th><math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/topos'>toposes</a><math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></th><th>locally presentable</th><th>loc finitely pres</th><th>localization theorem</th><th><a class='existingWikiWord' href='/nlab/show/diff/free+cocompletion'>free cocompletion</a></th><th>accessible</th></tr></thead><tbody><tr><td style='text-align: left;'><strong><a class='existingWikiWord' href='/nlab/show/diff/%280%2C1%29-category+theory'>(0,1)-category theory</a></strong></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/locale'>locales</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/suplattice'>suplattice</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/algebraic+lattice'>algebraic lattices</a></td><td style='text-align: left;'><a href='algebraic+lattice#RelationToLocallyFinitelyPresentableCategories'>Porst’s theorem</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/power+set'>powerset</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/partial+order'>poset</a></td></tr> <tr><td style='text-align: left;'><strong><a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a></strong></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+topos'>toposes</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+category'>locally presentable categories</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/locally+finitely+presentable+category'>locally finitely presentable categories</a></td><td style='text-align: left;'><a href='locally+presentable+category#AsLocalizationsOfPresheafCategories'>Gabriel–Ulmer’s theorem</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/category+of+presheaves'>presheaf category</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/accessible+category'>accessible categories</a></td></tr> <tr><td style='text-align: left;'><strong><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category theory</a></strong></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/model+topos'>model toposes</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/combinatorial+model+category'>combinatorial model categories</a></td><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Dugger%27s+theorem'>Dugger's theorem</a></td><td style='text-align: left;'>global <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+presheaves'>model structures on simplicial presheaves</a></td><td style='text-align: left;'>n/a</td></tr> <tr><td style='text-align: left;'><strong><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a></strong></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-toposes</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>locally presentable (∞,1)-categories</a></td><td style='text-align: left;' /><td style='text-align: left;'><a href='locally+presentable+infinity-category#Definition'>Simpson’s theorem</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-presheaves'>(∞,1)-presheaf (∞,1)-categories</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/accessible+%28infinity%2C1%29-category'>accessible (∞,1)-categories</a></td></tr> </tbody></table> <h2 id='references'>References</h2> <h3 id='ReferencesGeneral'>General</h3> <p>In retrospect, at least the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category+of+an+%28infinity%2C1%29-category'>homotopy categories</a> of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-toposes</a> have been known since</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Kenneth+Brown'>Kenneth Brown</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/BrownAHT'>Abstract homotopy theory and generalized sheaf cohomology</a></em>, Transactions of the American Mathematical Society, Vol. 186 (1973), 419-458 (<a href='http://www.jstor.org/stable/1996573'>jstor:1996573</a>).</li> </ul> <p>presented there via <a class='existingWikiWord' href='/nlab/show/diff/category+of+fibrant+objects'>categories of fibrant objects</a> among <a class='existingWikiWord' href='/nlab/show/diff/simplicial+presheaf'>simplicial presheaves</a>. The enhancement of this to <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model categories</a> <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+presheaves'>of simplicial presheaves</a> originates wit:h</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Andr%C3%A9+Joyal'>André Joyal</a>, Letter to <a class='existingWikiWord' href='/nlab/show/diff/Alexander+Grothendieck'>Alexander Grothendieck</a>, 11. 4. 1984, (<a href='http://webusers.imj-prg.fr/~georges.maltsiniotis/ps/lettreJoyal.pdf'>pdf scan</a>).</p> </li> <li id='JardineLecture'> <p><a class='existingWikiWord' href='/nlab/show/diff/John+Frederick+Jardine'>John F. Jardine</a>, <em>Simplicial presheaves</em>, Journal of Pure and Applied Algebra 47 (1987), 35-87 (<a href='https://core.ac.uk/download/pdf/82485559.pdf'>pdf</a>)</p> </li> </ul> <p>A more intrinsic characterization of these “<a class='existingWikiWord' href='/nlab/show/diff/model+topos'>model toposes</a>” (<a href='#Rezk10'>Rezk 2010</a>) as <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-toposes (the term seems to first appear here in <a href='#Simpson99'>Simpson 1999</a>) is due to:</p> <ul> <li id='Simpson99'><a class='existingWikiWord' href='/nlab/show/diff/Carlos+Simpson'>Carlos Simpson</a>, <em>A Giraud-type characterization of the simplicial categories associated to closed model categories as <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-pretopoi</em> (<a href='http://arxiv.org/abs/math/9903167'>arXiv:math/9903167</a>)</li> </ul> <p>The generalization of these <a class='existingWikiWord' href='/nlab/show/diff/model+topos'>model toposes</a> from 1-sites to <a class='existingWikiWord' href='/nlab/show/diff/sSet-site'>simplicial</a> <a class='existingWikiWord' href='/nlab/show/diff/model+site'>model sites</a> is due to</p> <ul> <li id='ToenVezzosi05'><a class='existingWikiWord' href='/nlab/show/diff/Bertrand+To%C3%ABn'>Bertrand Toën</a>, <a class='existingWikiWord' href='/nlab/show/diff/Gabriele+Vezzosi'>Gabriele Vezzosi</a>, <em>Homotopical Algebraic Geometry I: Topos theory</em>, Advances in Mathematics <strong>193</strong> 2 (2005) 257-372 [[arXiv:math.AG/0207028](http://arxiv.org/abs/math.AG/0207028), <a href='https://doi.org/10.1016/j.aim.2004.05.004'>doi:10.1016/j.aim.2004.05.004</a>]</li> </ul> <p>The term <em><math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-topos</em> is due to</p> <ul> <li id='Lurie03'><a class='existingWikiWord' href='/nlab/show/diff/Jacob+Lurie'>Jacob Lurie</a>, <em>On <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-Topoi</em> (<a href='https://arxiv.org/abs/math/0306109'>arXiv:math/0306109</a>)</li> </ul> <p>The term <em><a class='existingWikiWord' href='/nlab/show/diff/model+topos'>model topos</a></em> was later coined in:</p> <ul> <li id='Rezk10'><a class='existingWikiWord' href='/nlab/show/diff/Charles+Rezk'>Charles Rezk</a>, <em>Toposes and homotopy toposes</em>, 2010 (<a href='http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/Rezk_HomotopyToposes.pdf' title='pdf'>pdf</a>)</li> </ul> <p>A comprehensive conceptualization and discussion of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-toposes</a> is then due to</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Jacob+Lurie'>Jacob Lurie</a>, section 6.1 of: <em><a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>Higher Topos Theory</a></em>, Annals of Mathematics Studies 170, Princeton University Press 2009 (<a href='https://press.princeton.edu/titles/8957.html'>pup:8957</a>, <a href='https://www.math.ias.edu/~lurie/papers/HTT.pdf'>pdf</a>)</li> </ul> <p>building on <a href='#Rezk10'>Rezk 2010</a>. There is is also proven that the Brown-Joyal-Jardine-Toën-Vezzosi models indeed precisely model <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-stack <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_138' xmlns='http://www.w3.org/1998/Math/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>-toposes. Details on this relation are at <a class='existingWikiWord' href='/nlab/show/diff/presentations+of+%28infinity%2C1%29-sheaf+%28infinity%2C1%29-toposes'>models for ∞-stack (∞,1)-toposes</a>.</p> <p>Overview:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Denis-Charles+Cisinski'>Denis-Charles Cisinski</a>, <em>Catégories supérieures et théorie des topos</em>, Séminaire Bourbaki, 21.3.2015, <a href='http://www.math.univ-toulouse.fr/~dcisinsk/1097.pdf'>pdf</a>.</p> </li> <li id='Rezk19'> <p><a class='existingWikiWord' href='/nlab/show/diff/Charles+Rezk'>Charles Rezk</a>, <em>Lectures on Higher Topos Theory</em>, Leeds (2019) [[pdf](https://rezk.web.illinois.edu/leeds-lectures-2019.pdf), <a class='existingWikiWord' href='/nlab/files/RezkHigherToposTheory2019.pdf' title='pdf'>pdf</a>]</p> </li> </ul> <p>A useful collection of facts of <a class='existingWikiWord' href='/nlab/show/diff/simplicial+homotopy+theory'>simplicial homotopy theory</a> and <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos+theory'>(infinity,1)-topos theory</a> is in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Zhen+Lin+Low'>Zhen Lin Low</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Notes+on+homotopical+algebra'>Notes on homotopical algebra</a></em></li> </ul> <p>A quick introduction to the topic is in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Andr%C3%A9+Joyal'>André Joyal</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/A+crash+course+in+topos+theory+--+The+big+picture'>A crash course in topos theory -- The big picture</a></em>, lecture series at <a href='https://indico.math.cnrs.fr/event/747/'>Topos à l’IHES</a>, November 2015, Paris</li> </ul> <p>On <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_139' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-toposes <a class='existingWikiWord' href='/nlab/show/diff/category+object+in+an+%28infinity%2C1%29-category'>internal to</a> other <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_140' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-toposes;</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Louis+Martini'>Louis Martini</a>, <a class='existingWikiWord' href='/nlab/show/diff/Sebastian+Wolf'>Sebastian Wolf</a>, <em>Internal higher topos theory</em> [[arXiv:2303.06437](https://arxiv.org/abs/2303.06437)]</li> </ul> <h3 id='giraudrezklurie_axioms'>Giraud-Rezk-Lurie axioms</h3> <p>A discussion of the <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_141' xmlns='http://www.w3.org/1998/Math/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>-<a class='existingWikiWord' href='/nlab/show/diff/pullback-stable+colimit'>universal colimits</a> in terms of <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a> presentations is due to</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Charles+Rezk'>Charles Rezk</a>, <em>Fibrations and homotopy colimits of simplicial sheaves</em> (<a href='http://www.math.uiuc.edu/~rezk/rezk-sharp-maps.pdf'>pdf</a>)</li> </ul> <p>More on this with an eye on <a class='existingWikiWord' href='/nlab/show/diff/associated+infinity-bundle'>associated ∞-bundles</a> is in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Matthias+Wendt'>Matthias Wendt</a>, <em>Classifying spaces and fibrations of simplicial sheaves</em> (<a href='http://arxiv.org/abs/1009.2930'>arXiv</a>)</li> </ul> <h3 id='homotopy_type_theory'>Homotopy type theory</h3> <p>Proof that all <a class='existingWikiWord' href='/nlab/show/diff/infinity-stack'>∞-stack</a> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a> have <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presentations</a> by <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model categories</a> which interpret (provide <a class='existingWikiWord' href='/nlab/show/diff/categorical+semantics'>categorical semantics</a>) for <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a> with <a class='existingWikiWord' href='/nlab/show/diff/univalence+axiom'>univalent</a> <a class='existingWikiWord' href='/nlab/show/diff/type+universe'>type universes</a>:</p> <ul> <li id='Shulman19'><a class='existingWikiWord' href='/nlab/show/diff/Mike+Shulman'>Michael Shulman</a>, <em>All <math class='maruku-mathml' display='inline' id='mathml_f0784f67acc8cc6018975e8b5217d61a4a4aedcf_142' xmlns='http://www.w3.org/1998/Math/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>-toposes have strict univalent universes</em> (<a href='https://arxiv.org/abs/1904.07004'>arXiv:1904.07004</a>).</li> </ul> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on February 12, 2025 at 21:39:10. See the <a href="/nlab/history/%28infinity%2C1%29-topos" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/%28infinity%2C1%29-topos" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/9467/#Item_9">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/%28infinity%2C1%29-topos/88" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/%28infinity%2C1%29-topos" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/%28infinity%2C1%29-topos" accesskey="S" class="navlink" id="history" rel="nofollow">History (88 revisions)</a> <a href="/nlab/show/%28infinity%2C1%29-topos/cite" style="color: black">Cite</a> <a href="/nlab/print/%28infinity%2C1%29-topos" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/%28infinity%2C1%29-topos" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>