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href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> <h4 id="1topos_theory">(∞,1)-Topos theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a></strong></p> <h2 id="sidebar_background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf">(∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elementary+%28%E2%88%9E%2C1%29-topos">elementary (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">localization of an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+localization">topological localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercompletion">hypercompletion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>/<a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-topos">(n,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/n-topos">n-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/truncated">n-truncated object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected">n-connected object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">(1,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-presheaf">(2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-quasitopos">(∞,1)-quasitopos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+%28%E2%88%9E%2C1%29-presheaf">separated (∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+presheaf">separated presheaf</a></li> </ul> </li> <li> <p><span class="newWikiWord">(2,1)-quasitopos<a href="/nlab/new/%282%2C1%29-quasitopos">?</a></span></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+%282%2C1%29-presheaf">separated (2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-topos">(∞,2)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-topos">(∞,n)-topos</a></p> </li> </ul> <h2 id="sidebar_characterization">Characterization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object in an (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/effective+epimorphism">effective epimorphism</a></li> </ul> </li> </ul> <h2 id="sidebar_morphisms">Morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-geometric+morphism">(∞,1)-geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Topos">(∞,1)Topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lawvere+distribution">Lawvere distribution</a></p> </li> </ul> <h2 id="sidebar_extra">Extra stuff, structure and property</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+%28%E2%88%9E%2C1%29-topos">hypercomplete (∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+object">hypercomplete object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+theorem">Whitehead theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over-%28%E2%88%9E%2C1%29-topos">over-(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-localic+%28%E2%88%9E%2C1%29-topos">n-localic (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+n-connected+%28n%2C1%29-topos">locally n-connected (n,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/concrete+%28%E2%88%9E%2C1%29-sheaf">concrete (∞,1)-sheaf</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></p> </li> </ul> <h2 id="sidebar_models">Models</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+functors">model structure on functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+site">model site</a>/<a class="existingWikiWord" href="/nlab/show/sSet-site">sSet-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent+for+simplicial+presheaves">descent for simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Verity+on+descent+for+strict+omega-groupoid+valued+presheaves">descent for presheaves with values in strict ∞-groupoids</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_constructions">Constructions</h2> <p><strong>structures in a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/shape+of+an+%28%E2%88%9E%2C1%29-topos">shape</a> / <a class="existingWikiWord" href="/nlab/show/coshape+of+an+%28%E2%88%9E%2C1%29-topos">coshape</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a>/<a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical</a>/<a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric</a> homotopy groups</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Postnikov+tower+in+an+%28%E2%88%9E%2C1%29-category">Postnikov tower</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+tower+in+an+%28%E2%88%9E%2C1%29-topos">Whitehead tower</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory+in+an+%28%E2%88%9E%2C1%29-topos">rational homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+dimension">homotopy dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomological+dimension">cohomological dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covering+dimension">covering dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+dimension">Heyting dimension</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/%28infinity%2C1%29-topos+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="foundations">Foundations</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/foundations">foundations</a></strong></p> <h2 id="the_basis_of_it_all">The basis of it all</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mathematical+logic">mathematical logic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deduction+system">deduction system</a>, <a class="existingWikiWord" href="/nlab/show/natural+deduction">natural deduction</a>, <a class="existingWikiWord" href="/nlab/show/sequent+calculus">sequent calculus</a>, <a class="existingWikiWord" href="/nlab/show/lambda-calculus">lambda-calculus</a>, <a class="existingWikiWord" href="/nlab/show/judgment">judgment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a>, <a class="existingWikiWord" href="/nlab/show/simple+type+theory">simple type theory</a>, <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/collection">collection</a>, <a class="existingWikiWord" href="/nlab/show/object">object</a>, <a class="existingWikiWord" href="/nlab/show/type">type</a>, <a class="existingWikiWord" href="/nlab/show/term">term</a>, <a class="existingWikiWord" href="/nlab/show/set">set</a>, <a class="existingWikiWord" href="/nlab/show/element">element</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equality">equality</a>, <a class="existingWikiWord" href="/nlab/show/judgmental+equality">judgmental equality</a>, <a class="existingWikiWord" href="/nlab/show/typal+equality">typal equality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universe">universe</a>, <a class="existingWikiWord" href="/nlab/show/size+issues">size issues</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher-order+logic">higher-order logic</a></p> </li> </ul> <h2 id="set_theory"> Set theory</h2> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a></strong></p> <ul> <li>fundamentals of set theory <ul> <li><a class="existingWikiWord" href="/nlab/show/propositional+logic">propositional logic</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+logic">first-order logic</a></li> <li><a class="existingWikiWord" href="/nlab/show/typed+predicate+logic">typed predicate logic</a></li> <li><a class="existingWikiWord" href="/nlab/show/membership+relation">membership relation</a></li> <li><a class="existingWikiWord" href="/nlab/show/propositional+equality">propositional equality</a></li> <li><a class="existingWikiWord" href="/nlab/show/set">set</a>, <a class="existingWikiWord" href="/nlab/show/element">element</a>, <a class="existingWikiWord" href="/nlab/show/function">function</a>, <a class="existingWikiWord" href="/nlab/show/relation">relation</a></li> <li><a class="existingWikiWord" href="/nlab/show/universe">universe</a>, <a class="existingWikiWord" href="/nlab/show/small+set">small set</a>, <a class="existingWikiWord" href="/nlab/show/large+set">large set</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/material+set+theory">material set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/membership+relation">membership relation</a>, <a class="existingWikiWord" href="/nlab/show/propositional+equality">propositional equality</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+extensionality">axiom of extensionality</a></li> <li><a class="existingWikiWord" href="/nlab/show/pairing+structure">pairing structure</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+pairing">axiom of pairing</a></li> <li><a class="existingWikiWord" href="/nlab/show/union+structure">union structure</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+union">axiom of union</a></li> <li><a class="existingWikiWord" href="/nlab/show/powerset+structure">powerset structure</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+power+sets">axiom of power sets</a></li> <li><a class="existingWikiWord" href="/nlab/show/natural+numbers+structure">natural numbers structure</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+infinity">axiom of infinity</a></li> </ul> </li> <li>presentations of set theory <ul> <li><a class="existingWikiWord" href="/nlab/show/first-order+set+theory">first-order set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/unsorted+set+theory">unsorted set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/simply+sorted+set+theory">simply sorted set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/one-sorted+set+theory">one-sorted set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/two-sorted+set+theory">two-sorted set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/three-sorted+set+theory">three-sorted set theory</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/dependently+sorted+set+theory">dependently sorted set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/structurally+presented+set+theory">structurally presented set theory</a></li> </ul> </li> <li>structuralism in set theory <ul> <li><a class="existingWikiWord" href="/nlab/show/material+set+theory">material set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/ZFC">ZFC</a></li> <li><a class="existingWikiWord" href="/nlab/show/ZFA">ZFA</a></li> <li><a class="existingWikiWord" href="/nlab/show/Mostowski+set+theory">Mostowski set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/New+Foundations">New Foundations</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/structural+set+theory">structural set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/categorical+set+theory">categorical set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/ETCS">ETCS</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/fully+formal+ETCS">fully formal ETCS</a></li> <li><a class="existingWikiWord" href="/nlab/show/ETCS+with+elements">ETCS with elements</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+on+ETCS+I">Trimble on ETCS I</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+on+ETCS+II">Trimble on ETCS II</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+on+ETCS+III">Trimble on ETCS III</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/structural+ZFC">structural ZFC</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/allegorical+set+theory">allegorical set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/SEAR">SEAR</a></li> </ul> </li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/class-set+theory">class-set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/class">class</a>, <a class="existingWikiWord" href="/nlab/show/proper+class">proper class</a></li> <li><a class="existingWikiWord" href="/nlab/show/universal+class">universal class</a>, <a class="existingWikiWord" href="/nlab/show/universe">universe</a></li> <li><a class="existingWikiWord" href="/nlab/show/category+of+classes">category of classes</a></li> <li><a class="existingWikiWord" href="/nlab/show/category+with+class+structure">category with class structure</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/constructive+set+theory">constructive set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+set+theory">algebraic set theory</a></li> </ul> </div> <h2 id="foundational_axioms">Foundational axioms</h2> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/foundational+axiom">foundational</a> <a class="existingWikiWord" href="/nlab/show/axioms">axioms</a></strong></p> <ul> <li> <p>basic constructions:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+cartesian+products">axiom of cartesian products</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+disjoint+unions">axiom of disjoint unions</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+the+empty+set">axiom of the empty set</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+fullness">axiom of fullness</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+function+sets">axiom of function sets</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+power+sets">axiom of power sets</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+quotient+sets">axiom of quotient sets</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/material+set+theory">material axioms</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+extensionality">axiom of extensionality</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+foundation">axiom of foundation</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+anti-foundation">axiom of anti-foundation</a></li> <li><a class="existingWikiWord" href="/nlab/show/Mostowski%27s+axiom">Mostowski's axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+pairing">axiom of pairing</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+transitive+closure">axiom of transitive closure</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+union">axiom of union</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/structural+set+theory">structural axioms</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+materialization">axiom of materialization</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+theory">type theoretic axioms</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axioms+of+set+truncation">axioms of set truncation</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/uniqueness+of+identity+proofs">uniqueness of identity proofs</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+K">axiom K</a></li> <li><a class="existingWikiWord" href="/nlab/show/boundary+separation">boundary separation</a></li> <li><a class="existingWikiWord" href="/nlab/show/equality+reflection">equality reflection</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+circle+type+localization">axiom of circle type localization</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theoretic axioms</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/univalence+axiom">univalence axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+principle">Whitehead's principle</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/axioms+of+choice">axioms of choice</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+countable+choice">axiom of countable choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+dependent+choice">axiom of dependent choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+excluded+middle">axiom of excluded middle</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+existence">axiom of existence</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+multiple+choice">axiom of multiple choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/Markov%27s+axiom">Markov's axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/presentation+axiom">presentation axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/small+cardinality+selection+axiom">small cardinality selection axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+small+violations+of+choice">axiom of small violations of choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+weakly+initial+sets+of+covers">axiom of weakly initial sets of covers</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/large+cardinal+axioms">large cardinal axioms</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+infinity">axiom of infinity</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+universes">axiom of universes</a></li> <li><a class="existingWikiWord" href="/nlab/show/regular+extension+axiom">regular extension axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/inaccessible+cardinal">inaccessible cardinal</a></li> <li><a class="existingWikiWord" href="/nlab/show/measurable+cardinal">measurable cardinal</a></li> <li><a class="existingWikiWord" href="/nlab/show/elementary+embedding">elementary embedding</a></li> <li><a class="existingWikiWord" href="/nlab/show/supercompact+cardinal">supercompact cardinal</a></li> <li><a class="existingWikiWord" href="/nlab/show/Vop%C4%9Bnka%27s+principle">Vopěnka's principle</a></li> </ul> </li> <li> <p>strong axioms</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+separation">axiom of separation</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+replacement">axiom of replacement</a></li> </ul> </li> <li> <p>further</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/reflection+principle">reflection principle</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/axiom+of+inequality+spaces">axiom of inequality spaces</a></p> </li> </ul> </div> <h2 id="removing_axioms">Removing axioms</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a></li> <li><a class="existingWikiWord" href="/nlab/show/predicative+mathematics">predicative mathematics</a></li> </ul> <div> <p> <a href="/nlab/edit/foundations+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#classical_case'>Classical case</a></li> <li><a href='#simplicial_version'>Simplicial version</a></li> <li><a href='#equivariant_version'>Equivariant version</a></li> <li><a href='#in_general_toposes'>In general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes</a></li> <li><a href='#in_homotopy_type_theory'>In homotopy type theory</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="classical_case">Classical case</h2> <p>The classical <strong>Whitehead theorem</strong> asserts that:</p> <p> <div class='num_theorem'> <h6>Theorem</h6> <p>(<a href="#Whitehead49">Whitehead 1949</a>) Every <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> between <a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a> is a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>.</p> </div> </p> <p>(See also the discussion at <em><a class="existingWikiWord" href="/nlab/show/m-cofibrant+space">m-cofibrant space</a></em>).</p> <p>Using the <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem this may be reformulated:</p> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>In the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> every <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> is a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>.</p> </div> <h2 id="simplicial_version">Simplicial version</h2> <p> <div class='num_theorem'> <h6>Theorem</h6> <p>A <a class="existingWikiWord" href="/nlab/show/simplicial+map">simplicial map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X\to Y</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a> is a <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+equivalence">simplicial homotopy equivalence</a> if and only if for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> that make the following square commute</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>∂</mo><msup><mi>Δ</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>→</mo><mi>a</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mi>ι</mi></msup></mtd> <mtd><msup><mrow></mrow> <mrow><mo>∃</mo><mi>d</mi></mrow></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>f</mi></msup></mtd></mtr> <mtr><mtd><msup><mi>Δ</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>→</mo><mi>b</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \partial\Delta^n&\stackrel{a}{\to}&X\\ \downarrow^\iota&{}^{\exists d}\nearrow&\downarrow^f\\ \Delta^n &\stackrel{b}{\to}&Y\\ } </annotation></semantics></math></div> <p>there is a diagonal arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> that makes the upper triangle commutative and the lower triangle commutative up to a homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo lspace="verythinmathspace">:</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo>×</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">h\colon \Delta^1\times\Delta^n\to Y</annotation></semantics></math> that is constant on the boundary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>1</mn></msup><mo>×</mo><mo>∂</mo><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Delta^1\times\partial\Delta^n</annotation></semantics></math>.</p> </div> </p> <p>Of course, this statement can be reformulated using homotopy groups like the version for topological spaces, but the above statement is more practical.</p> <p> <div class='num_remark'> <h6>Remark</h6> <p>In the above criterion, the boundary inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∂</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\partial\Delta^n\to\Delta^n</annotation></semantics></math></div> <p>can be replaced by any weakly equivalent <a class="existingWikiWord" href="/nlab/show/cofibration">cofibration</a>.</p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is not a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>, one can formulate a similar criterion using <a class="existingWikiWord" href="/nlab/show/barycentric+subdivisions">barycentric subdivisions</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\partial\Delta^n</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Delta^n</annotation></semantics></math>. A <a class="existingWikiWord" href="/nlab/show/simplicial+map">simplicial map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X\to Y</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> if and only if for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">k\ge0</annotation></semantics></math> and for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> that make the following square commute</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>Sd</mi> <mi>k</mi></msup><mo>∂</mo><msup><mi>Δ</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>→</mo><mi>a</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mrow><msup><mi>Sd</mi> <mi>k</mi></msup><mi>ι</mi></mrow></msup></mtd> <mtd><msup><mrow></mrow> <mrow><mo>∃</mo><mi>d</mi></mrow></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>f</mi></msup></mtd></mtr> <mtr><mtd><msup><mi>Sd</mi> <mi>k</mi></msup><msup><mi>Δ</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>→</mo><mi>b</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ Sd^k \partial\Delta^n&\stackrel{a}{\to}&X\\ \downarrow^{Sd^k \iota}&{}^{\exists d}\nearrow&\downarrow^f\\ Sd^k \Delta^n &\stackrel{b}{\to}&Y\\ } </annotation></semantics></math></div> <p>there is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo>≥</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">l\ge k</annotation></semantics></math> such that in the outer rectangle in the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>Sd</mi> <mi>l</mi></msup><mo>∂</mo><msup><mi>Δ</mi> <mi>n</mi></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>Sd</mi> <mi>k</mi></msup><mo>∂</mo><msup><mi>Δ</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>→</mo><mi>a</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mrow><msup><mi>Sd</mi> <mi>l</mi></msup><mi>ι</mi></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msup><mi>Sd</mi> <mi>k</mi></msup><mi>ι</mi></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>f</mi></msup></mtd></mtr> <mtr><mtd><msup><mi>Sd</mi> <mi>l</mi></msup><msup><mi>Δ</mi> <mi>n</mi></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>Sd</mi> <mi>k</mi></msup><msup><mi>Δ</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>→</mo><mi>b</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ Sd^l \partial\Delta^n&\to&Sd^k \partial\Delta^n&\stackrel{a}{\to}&X\\ \downarrow^{Sd^l \iota}&&\downarrow^{Sd^k \iota}&&\downarrow^f\\ Sd^l \Delta^n &\to&Sd^k \Delta^n &\stackrel{b}{\to}&Y\\ } </annotation></semantics></math></div> <p>we can find a diagonal arrow</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo lspace="verythinmathspace">:</mo><msup><mi>Sd</mi> <mi>l</mi></msup><msup><mi>Δ</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">d\colon Sd^l \Delta^n \to X</annotation></semantics></math></div> <p>that makes the upper triangle in the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>Sd</mi> <mi>l</mi></msup><mo>∂</mo><msup><mi>Δ</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>→</mo><mi>a</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mrow><msup><mi>Sd</mi> <mi>l</mi></msup><mi>ι</mi></mrow></msup></mtd> <mtd><msup><mrow></mrow> <mrow><mo>∃</mo><mi>d</mi></mrow></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>f</mi></msup></mtd></mtr> <mtr><mtd><msup><mi>Sd</mi> <mi>l</mi></msup><msup><mi>Δ</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>→</mo><mi>b</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ Sd^l \partial\Delta^n&\stackrel{a}{\to}&X\\ \downarrow^{Sd^l \iota}&{}^{\exists d}\nearrow&\downarrow^f\\ Sd^l \Delta^n &\stackrel{b}{\to}&Y\\ } </annotation></semantics></math></div> <p>commutative and the lower triangle commutative up to a homotopy</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo lspace="verythinmathspace">:</mo><msup><mi>Sd</mi> <mi>l</mi></msup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo>×</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">h\colon Sd^l(\Delta^1\times\Delta^n)\to Y</annotation></semantics></math></div> <p>that is constant on the boundary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Sd</mi> <mi>l</mi></msup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo>×</mo><mo>∂</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sd^l(\Delta^1\times\partial\Delta^n)</annotation></semantics></math>.</p> </div> </p> <h2 id="equivariant_version">Equivariant version</h2> <p>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a> the statement is that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-homotopy equivalences between <a class="existingWikiWord" href="/nlab/show/G-CW+complexes">G-CW complexes</a> are equivalent to maps that are weak homotopy equivalences on <a class="existingWikiWord" href="/nlab/show/fixed+point">fixed point</a> spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>H</mi></msup></mrow><annotation encoding="application/x-tex">H^H</annotation></semantics></math> for all <a class="existingWikiWord" href="/nlab/show/closed+subset">closed</a> <a class="existingWikiWord" href="/nlab/show/subgroups">subgroups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>⊂</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \subset G</annotation></semantics></math> (e.g. <a href="#GreenleesMay95">Greenlees-May 95, theorem 2.4</a>). See at <em><a class="existingWikiWord" href="/nlab/show/equivariant+Whitehead+theorem">equivariant Whitehead theorem</a></em>.</p> <h2 id="in_general_toposes">In general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes</h2> <p>There is a notion of <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> for objects in every <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>, as described at <a class="existingWikiWord" href="/nlab/show/homotopy+group+%28of+an+%E2%88%9E-stack%29">homotopy group (of an ∞-stack)</a>. Accordingly, there is a notion of <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> in every <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> and hence an analog of the statement of Whiteheads theorem. One finds that</p> <p><strong>Warning</strong> Whitehead’s theorem <strong>fails</strong> for general <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-toposes">(∞,1)-toposes</a> and non-<a class="existingWikiWord" href="/nlab/show/truncated+object+in+an+%28infinity%2C1%29-category">truncated</a> objects.</p> <p>The <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es in which the Whitehead theorem does hold are the <a class="existingWikiWord" href="/nlab/show/hypercomplete+%28%E2%88%9E%2C1%29-topos">hypercomplete (∞,1)-topos</a>es. These are precisely the ones that are <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presented</a> by a local <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a>.</p> <p>For instance the hypercomplete <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos <a class="existingWikiWord" href="/nlab/show/Top">Top</a> is presented by the model structure on simplicial presheaves on the point, namely the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a>.</p> <h2 id="in_homotopy_type_theory">In homotopy type theory</h2> <p>Since <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> admits models in <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-toposes">(∞,1)-toposes</a> (and in particular in non-hypercomplete ones), Whitehead’s theorem is not provable when regarded as a statement about types in homotopy type theory. From this perspective, the truth of Whitehead’s theorem is a <a class="existingWikiWord" href="/nlab/show/foundational+axiom">foundational axiom</a> that may be regarded as a “classicality” property, akin to <a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a> or the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> — we call it <strong>Whitehead’s principle</strong> (not to be confused with <span class="newWikiWord">Whitehead's problem<a href="/nlab/new/Whitehead%27s+problem">?</a></span>, another statement that is independent of the usual axioms of set theory).</p> <p>Whitehead’s principle does hold, however, for maps between <a class="existingWikiWord" href="/nlab/show/homotopy+n-types">homotopy n-types</a> for any finite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>; this is provable in homotopy type theory by <a class="existingWikiWord" href="/nlab/show/induction">induction</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mod+p+Whitehead+theorem">mod p Whitehead theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+Whitehead+theorem">equivariant Whitehead theorem</a></p> </li> <li> <p><a href="G-spectrum#EquivariantWhitehead">equivariant stable Whitehead theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CW-approximation+theorem">CW-approximation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cellular+approximation+theorem">cellular approximation theorem</a></p> </li> </ul> <h2 id="references">References</h2> <p>The original theorem for maps between <a class="existingWikiWord" href="/nlab/show/CW+complexes">CW complexes</a>:</p> <ul> <li id="Whitehead49"><a class="existingWikiWord" href="/nlab/show/J.+H.+C.+Whitehead">J. H. C. Whitehead</a>, <em>Combinatorial homotopy. I</em>, Bulletin of the American Mathematical Society <strong>55</strong> 3 (1949) 213–245 [<a href="http://dx.doi.org/10.1090/s0002-9904-1949-09175-9">doi:10.1090/s0002-9904-1949-09175-9</a>, <a href="https://www.ams.org/journals/bull/1949-55-03/S0002-9904-1949-09175-9/S0002-9904-1949-09175-9.pdf">pdf</a>]</li> </ul> <p>The version in <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial homotopy theory</a> is due to:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+M.+Kan">Daniel M. Kan</a>, Theorem 7.2 in: <em>On c.s.s. categories</em>, Boletín de la Sociedad Matemática Mexicana <strong>2</strong> (1957) 82–94 [<a href="https://www.boletin.math.org.mx/pdf/2/2/BSMM(2).2.82-94.pdf">pdf</a>, <a href="https://dmitripavlov.org/scans/kan-on-css-categories.pdf">pdf</a>]</li> </ul> <p>Review:</p> <ul> <li id="Bredon93"> <p><a class="existingWikiWord" href="/nlab/show/Glen+Bredon">Glen Bredon</a>, Cor. 11.14 in: <em>Topology and Geometry</em>, Graduate texts in mathematics <strong>139</strong>, Springer 1993 (<a href="https://link.springer.com/book/10.1007/978-1-4757-6848-0">doi:10.1007/978-1-4757-6848-0</a>, <a href="http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf">pdf</a>)</p> </li> <li id="ElmendorfKrizMay95"> <p><a class="existingWikiWord" href="/nlab/show/Anthony+Elmendorf">Anthony Elmendorf</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Kriz">Igor Kriz</a>, <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, section 1 of <em><a class="existingWikiWord" href="/nlab/show/Modern+foundations+for+stable+homotopy+theory">Modern foundations for stable homotopy theory</a></em>, in <a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a> (ed.), <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Algebraic+Topology">Handbook of Algebraic Topology</a></em>, 1995 Amsterdam: North-Holland, pp. 213–253, (<a href="http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Marcelo+Aguilar">Marcelo Aguilar</a>, <a class="existingWikiWord" href="/nlab/show/Samuel+Gitler">Samuel Gitler</a>, <a class="existingWikiWord" href="/nlab/show/Carlos+Prieto">Carlos Prieto</a>, Thm. 6.3.31 in: <em>Algebraic topology from a homotopical viewpoint</em>, Springer (2008) (<a href="https://link.springer.com/book/10.1007/b97586">doi:10.1007/b97586</a>)</p> </li> </ul> <p>Discussion of the <a class="existingWikiWord" href="/nlab/show/equivariant+Whitehead+theorem">equivariant Whitehead theorem</a>:</p> <ul> <li id="GreenleesMay95"><a class="existingWikiWord" href="/nlab/show/John+Greenlees">John Greenlees</a>, <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <em>Equivariant stable homotopy theory</em>, in I.M. James (ed.), <em>Handbook of Algebraic Topology</em> , pp. 279-325. 1995. (<a href="http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf">pdf</a>)</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>∞</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">(\infty,1)</annotation> </semantics> </math>-topos</a> theoretic version:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, section 6.5 of <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></li> </ul> <p>The analogous formulation in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>:</p> <ul> <li id="UFP13"><a class="existingWikiWord" href="/nlab/show/Univalent+Foundations+Project">Univalent Foundations Project</a>, §8.8 in: <em><a class="existingWikiWord" href="/nlab/show/Homotopy+Type+Theory+--+Univalent+Foundations+of+Mathematics">Homotopy Type Theory – Univalent Foundations of Mathematics</a></em> (2013) [<a href="http://homotopytypetheory.org/book/">web</a>, <a href="http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf">pdf</a>]</li> </ul> <p>Corresponding formalization in <a class="existingWikiWord" href="/nlab/show/Agda">Agda</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dan+Licata">Dan Licata</a>, <em><a href="https://github.com/dlicata335/hott-agda/blob/master/homotopy/Whitehead.agda">Whitehead.agda</a></em></li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/foundational+axiom">foundational axiom</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on June 23, 2024 at 16:08:15. 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