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href="/nlab/show/differential+topology">differential topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></em> and <em><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></em></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology">Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>, <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+for+the+topology">base for the topology</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood+base">neighbourhood base</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finer+topology">finer/coarser topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a>, <a class="existingWikiWord" href="/nlab/show/topological+interior">interior</a>, <a class="existingWikiWord" href="/nlab/show/topological+boundary">boundary</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separation+axiom">separation</a>, <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sobriety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/uniformly+continuous+function">uniformly continuous function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+embedding">embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+map">open map</a>, <a class="existingWikiWord" href="/nlab/show/closed+map">closed map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a>, <a class="existingWikiWord" href="/nlab/show/sub-net">sub-net</a>, <a class="existingWikiWord" href="/nlab/show/filter">filter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/convergence">convergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a><a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href="Top#UniversalConstructions">Universal constructions</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a>, <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>,</p> </li> <li> <p>fiber space, <a class="existingWikiWord" href="/nlab/show/space+attachment">space attachment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product+space">product space</a>, <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+telescope">mapping telescope</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colimits+of+normal+spaces">colimits of normal spaces</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>, <a class="existingWikiWord" href="/nlab/show/metrisable+space">metrisable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+space">Kolmogorov space</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, <a class="existingWikiWord" href="/nlab/show/regular+space">regular space</a>, <a class="existingWikiWord" href="/nlab/show/normal+space">normal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sober+space">sober space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>, <a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a></p> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+topological+space">sequentially compact</a>, <a class="existingWikiWord" href="/nlab/show/countably+compact+topological+space">countably compact</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>, <a class="existingWikiWord" href="/nlab/show/sigma-compact+topological+space">sigma-compact</a>, <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a>, <a class="existingWikiWord" href="/nlab/show/countably+paracompact+topological+space">countably paracompact</a>, <a class="existingWikiWord" href="/nlab/show/strongly+compact+topological+space">strongly compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+space">second-countable space</a>, <a class="existingWikiWord" href="/nlab/show/first-countable+space">first-countable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a>, <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+space">connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+connected+space">locally connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simply-connected+space">simply-connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+simply-connected+space">locally simply-connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a>, <a class="existingWikiWord" href="/nlab/show/point+space">point space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+space">codiscrete space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/order+topology">order topology</a>, <a class="existingWikiWord" href="/nlab/show/specialization+topology">specialization topology</a>, <a class="existingWikiWord" href="/nlab/show/Scott+topology">Scott topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, <a class="existingWikiWord" href="/nlab/show/plane">plane</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>, <a class="existingWikiWord" href="/nlab/show/cone">cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, <a class="existingWikiWord" href="/nlab/show/ball">ball</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle">circle</a>, <a class="existingWikiWord" href="/nlab/show/torus">torus</a>, <a class="existingWikiWord" href="/nlab/show/annulus">annulus</a>, <a class="existingWikiWord" href="/nlab/show/Moebius+strip">Moebius strip</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polytope">polytope</a>, <a class="existingWikiWord" href="/nlab/show/polyhedron">polyhedron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+space">projective space</a> (<a class="existingWikiWord" href="/nlab/show/real+projective+space">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configuration space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path">path</a>, <a class="existingWikiWord" href="/nlab/show/loop">loop</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>: <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>, <a class="existingWikiWord" href="/nlab/show/topology+of+uniform+convergence">topology of uniform convergence</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/path+space">path space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cantor+space">Cantor space</a>, <a class="existingWikiWord" href="/nlab/show/Mandelbrot+space">Mandelbrot space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+curve">Peano curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+with+two+origins">line with two origins</a>, <a class="existingWikiWord" href="/nlab/show/long+line">long line</a>, <a class="existingWikiWord" href="/nlab/show/Sorgenfrey+line">Sorgenfrey line</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-topology">K-topology</a>, <a class="existingWikiWord" href="/nlab/show/Dowker+space">Dowker space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Warsaw+circle">Warsaw circle</a>, <a class="existingWikiWord" href="/nlab/show/Hawaiian+earring+space">Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces+are+sober">Hausdorff spaces are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces">closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact">open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff">quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lebesgue+number+lemma">Lebesgue number lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces">sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+totally+bounded">sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous">continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+are+normal">paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity">paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+injections+are+embeddings">closed injections are embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+maps+to+locally+compact+spaces+are+closed">proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings">injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact">locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+second-countable+spaces+are+sigma-compact">locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+regular+spaces+are+paracompact">second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CW-complexes+are+paracompact+Hausdorff+spaces">CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urysohn%27s+lemma">Urysohn's lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tychonoff+theorem">Tychonoff theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tube+lemma">tube lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael%27s+theorem">Michael's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brouwer%27s+fixed+point+theorem">Brouwer's fixed point theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jordan+curve+theorem">Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <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/homotopy+group">homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</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/Freudenthal+suspension+theorem">Freudenthal suspension 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/homotopy+extension+property">homotopy extension property</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+cofiber+sequence">cofiber sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+category">Strøm model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></p> </li> </ul> </div></div> <h4 id="model_category_theory">Model category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></strong>, <a class="existingWikiWord" href="/nlab/show/model+%28infinity%2C1%29-category">model <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-category</a></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/relative+category">relative category</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration">fibration</a>, <a class="existingWikiWord" href="/nlab/show/cofibration">cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+factorization+system">weak factorization system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/resolution">resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+%28as+an+operation%29">homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">of a model category</a></p> </li> </ul> <p><strong>Morphisms</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+bifunctor">Quillen bifunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a></p> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> <p><a class="existingWikiWord" href="/nlab/show/homotopy+weighted+colimit">homotopy weighted (co)limit</a></p> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coend">homotopy (co)end</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield-Kan+map">Bousfield-Kan map</a></p> </li> </ul> <p><strong>Refinements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/monoidal+Quillen+adjunction">monoidal Quillen adjunction</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/enriched+Quillen+adjunction">enriched Quillen adjunction</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+enriched+model+category">monoidal enriched model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+Quillen+adjunction">simplicial Quillen adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+monoidal+model+category">simplicial monoidal model category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cellular+model+category">cellular model category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+model+category">algebraic model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+model+category">compactly generated model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+model+category">proper model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+model+category">cartesian closed model category</a>, <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+model+category">locally cartesian closed model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> </ul> <p><strong>Producing new model structures</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/global+model+structures+on+functor+categories">on functor categories (global)</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+an+overcategory">on slice categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">Bousfield localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebraic+fibrant+objects">on algebraic fibrant objects</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction+for+model+categories">Grothendieck construction for model categories</a></p> </li> </ul> <p><strong>Presentation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</strong></p> <ul> <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/simplicial+localization">simplicial localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categorical+hom-space">(∞,1)-categorical hom-space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable (∞,1)-category</a></p> </li> </ul> <p><strong>Model structures</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cisinski+model+structure">Cisinski model structure</a></li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</em></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+%E2%88%9E-groupoids">for ∞-groupoids</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+spaces">on topological spaces</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+compactly+generated+topological+spaces">on compactly generated spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+Delta-generated+topological+spaces">on Delta-generated spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+diffeological+spaces">on diffeological spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+structure">Strøm model structure</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thomason+model+structure">Thomason model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+over+a+test+category">model structure on presheaves over a test category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">on simplicial sets</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+semi-simplicial+sets">on semi-simplicial sets</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/constructive+model+structure+on+simplicial+sets">constructive model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+right+fibrations">for right/left fibrations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groupoids">model structure on simplicial groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cubical+sets">on cubical sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+%E2%88%9E-groupoids">on strict ∞-groupoids</a>, <a class="existingWikiWord" href="/nlab/show/natural+model+structure+on+groupoids">on groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">on chain complexes</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+abelian+groups">model structure on cosimplicial abelian groups</a></p> <p>related by the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+simplicial+sets">model structure on cosimplicial simplicial sets</a></p> </li> </ul> <p><em>for equivariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fine+model+structure+on+topological+G-spaces">fine model structure on topological G-spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coarse+model+structure+on+topological+G-spaces">coarse model structure on topological G-spaces</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/Borel+model+structure">Borel model structure</a>)</p> </li> </ul> <p><em>for rational <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</em></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dgc-algebras">model structure on dgc-algebras</a></li> </ul> <p><em>for rational equivariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+chain+complexes">model structure on equivariant chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+dgc-algebras">model structure on equivariant dgc-algebras</a></p> </li> </ul> <p><em>for <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>-groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+n-groupoids">for n-groupoids</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+for+n-groupoids">for n-types</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+groupoids">for 1-groupoids</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groups</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groups">model structure on simplicial groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+reduced+simplicial+sets">model structure on reduced simplicial sets</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras</em></p> <p><em>general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+monoids">on monoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">on simplicial T-algebras</a>, on <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a>s</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+a+monad">on algebas over a monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">on algebras over an operad</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad">on modules over an algebra over an operad</a></p> </li> </ul> <p><em>specific <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">model structure on differential-graded commutative algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure on differential graded-commutative superalgebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras+over+an+operad">on dg-algebras over an operad</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">on dg-algebras</a> and on <a class="existingWikiWord" href="/nlab/show/simplicial+ring">on simplicial rings</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+rings">on cosimplicial rings</a></p> <p>related by the <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-modules">model structure on dg-modules</a></p> </li> </ul> <p><em>for stable/spectrum objects</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+spectra">model structure on spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+ring+spectra">model structure on ring spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+parameterized+spectra">model structure on parameterized spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+spectra">model structure on presheaves of spectra</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+categories+with+weak+equivalences">on categories with weak equivalences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+quasi-categories">Joyal model for quasi-categories</a> (and its <a class="existingWikiWord" href="/nlab/show/model+structure+for+cubical+quasicategories">cubical version</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-categories">on sSet-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+complete+Segal+spaces">for complete Segal spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">for Cartesian fibrations</a></p> </li> </ul> <p><em>for stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</em></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-categories">on dg-categories</a></li> </ul> <p><em>for <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>-operads</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">on operads</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+Segal+operads">for Segal operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">on algebras over an operad</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad">on modules over an algebra over an operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dendroidal+sets">on dendroidal sets</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+complete+Segal+spaces">for dendroidal complete Segal spaces</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+Cartesian+fibrations">for dendroidal Cartesian fibrations</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,r)</annotation></semantics></math>-categories</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Theta+space">for (n,r)-categories as ∞-spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+weak+complicial+sets">for weak ∞-categories as weak complicial sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cellular+sets">on cellular sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+model+structure">on higher categories in general</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+%E2%88%9E-categories">on strict ∞-categories</a></p> </li> </ul> <p><em>for <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>-sheaves / <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+homotopical+presheaves">on homotopical presheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">on simplicial presheaves</a></p> <p><a class="existingWikiWord" href="/nlab/show/global+model+structure+on+simplicial+presheaves">global model structure</a>/<a class="existingWikiWord" href="/nlab/show/Cech+model+structure+on+simplicial+presheaves">Cech model structure</a>/<a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">local model structure</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sheaves">on simplicial sheaves</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+simplicial+groupoids">on presheaves of simplicial groupoids</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-enriched+presheaves">on sSet-enriched presheaves</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+%282%2C1%29-sheaves">model structure for (2,1)-sheaves</a>/for stacks</p> </li> </ul> </div></div> <h4 id="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" 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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#background_on_combinatorial_topology'>Background on combinatorial topology</a></li> <ul> <li><a href='#simplicial_sets'>Simplicial sets</a></li> <li><a href='#simplicial_homotopy'>Simplicial homotopy</a></li> <li><a href='#kan_fibrations'>Kan fibrations</a></li> <li><a href='#geometric_realization'>Geometric realization</a></li> </ul> <li><a href='#TheClassicalModelStructure'>The classical model structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math></a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#basic_properties'>Basic properties</a></li> <li><a href='#Properness'>Properness</a></li> <li><a href='#quillen_equivalence_with_'>Quillen equivalence with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math></a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#classical_model_structure_on_simplicial_sets'>Classical model structure on simplicial sets</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>classical model structure on simplicial sets</em> or <em>Kan-Quillen model structure</em> , <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math> (<a href="#Quillen67">Quillen 67, II.3</a>) is a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure on the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> which represents the standard classical <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>.</p> <p>Its <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> are the <a class="existingWikiWord" href="/nlab/show/simplicial+weak+equivalences">simplicial weak equivalences</a> (<a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> on <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+groups">simplicial homotopy groups</a>), its <a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a> are the <a class="existingWikiWord" href="/nlab/show/Kan+fibrations">Kan fibrations</a> and its <a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a> are the <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> (degreewise injections).</p> <p>The <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a>/<a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> <a class="existingWikiWord" href="/nlab/show/nerve+and+realization">adjunction</a> constitutes a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a>. This is sometimes called part of the statement of the <em><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a></em> <a href="homotopy+hypothesis#ForKanComplexes">for Kan complexes</a>. In the language of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a> this means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math> both are <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentations</a> of 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> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a>.</p> <p>There are also other model structures on <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> itself, see at <em><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a></em> for more. This entry here focuses on just the standard classical model structure.</p> <h2 id="background_on_combinatorial_topology">Background on combinatorial topology</h2> <p>This section reviews basics of the theory of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> (the modern version of the original “combinatorial topology”) necessary to define, verify and analyse the classical model category structure on simplicial sets, <a href="#TheClassicalModelStructure">below</a>. See also at <em><a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial homotopy theory</a></em>.</p> <h3 id="simplicial_sets">Simplicial sets</h3> <p>The concept of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> is secretly well familiar already in basic <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>: it reflects just the abstract structure carried by the <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complexes">singular simplicial complexes</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, as in the definition of <a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a> and <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a>.</p> <p>Conversely, every simplicial set may be <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometrically realized</a> as a topological space. These two <a class="existingWikiWord" href="/nlab/show/adjoint">adjoint</a> operations turn out to exhibit the homotopy theory of simplicial sets as being equivalent (<a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a>) to the homotopy theory of topological spaces. For some purposes, working in <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial homotopy theory</a> is preferable over working with topological homotopy theory.</p> <div class="num_defn" id="TopologicalSimplex"> <h6 id="definition">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, the <strong><a href="simplex#TopologicalSimplex">topological n-simplex</a></strong> is, up to <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a>, the <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> whose underlying set is the subset</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>n</mi></msup><mo>≔</mo><mo stretchy="false">{</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>∈</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">|</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mi>n</mi></munderover><msub><mi>x</mi> <mi>i</mi></msub><mo>=</mo><mn>1</mn><mspace width="thickmathspace"></mspace><mi>and</mi><mspace width="thickmathspace"></mspace><mo>∀</mo><mi>i</mi><mo>.</mo><msub><mi>x</mi> <mi>i</mi></msub><mo>≥</mo><mn>0</mn><mo stretchy="false">}</mo><mo>⊂</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> \Delta^n \coloneqq \{ \vec x \in \mathbb{R}^{n+1} | \sum_{i = 0 }^n x_i = 1 \; and \; \forall i . x_i \geq 0 \} \subset \mathbb{R}^{n+1} </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n+1}</annotation></semantics></math>, and whose topology is the <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a> induces from the canonical topology in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n+1}</annotation></semantics></math>.</p> </div> <div class="num_example"> <h6 id="example">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math> this is the <a class="existingWikiWord" href="/nlab/show/point">point</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>0</mn></msup><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\Delta^0 = *</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math> this is the standard <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a> <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 stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta^1 = [0,1]</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math> this is the filled triangle.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n = 3</annotation></semantics></math> this is the filled tetrahedron.</p> </div> <div class="num_defn" id="FaceInclusionInBarycentricCoords"> <h6 id="definition_2">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">n</mo><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\n \geq 1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0 \leq k \leq n</annotation></semantics></math>, the <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>th <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-face (inclusion)</strong> of the topological <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>-simplex, def. <a class="maruku-ref" href="#TopologicalSimplex"></a>, is the subspace inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>k</mi></msub><mo>:</mo><msup><mi>Δ</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>↪</mo><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> \delta_k : \Delta^{n-1} \hookrightarrow \Delta^n </annotation></semantics></math></div> <p>induced under the coordinate presentation of def. <a class="maruku-ref" href="#TopologicalSimplex"></a>, by the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>↪</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> \mathbb{R}^n \hookrightarrow \mathbb{R}^{n+1} </annotation></semantics></math></div> <p>which “omits” the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>th canonical coordinate:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mn>0</mn><mo>,</mo><msub><mi>x</mi> <mi>k</mi></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (x_0, \cdots , x_{n-1}) \mapsto (x_0, \cdots, x_{k-1} , 0 , x_{k}, \cdots, x_n) \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example_2">Example</h6> <p>The inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mn>0</mn></msub><mo>:</mo><msup><mi>Δ</mi> <mn>0</mn></msup><mo>→</mo><msup><mi>Δ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex"> \delta_0 : \Delta^0 \to \Delta^1 </annotation></semantics></math></div> <p>is the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo><mo>↪</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \{1\} \hookrightarrow [0,1] </annotation></semantics></math></div> <p>of the “right” end of the standard interval. The other inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mn>1</mn></msub><mo>:</mo><msup><mi>Δ</mi> <mn>0</mn></msup><mo>→</mo><msup><mi>Δ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex"> \delta_1 : \Delta^0 \to \Delta^1 </annotation></semantics></math></div> <p>is that of the “left” end <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo>↪</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\{0\} \hookrightarrow [0,1]</annotation></semantics></math>.</p> </div> <p><img src="http://ncatlab.org/nlab/files/faceanddegeneracymaps.jpg" width="500" /></p> <p>(graphics taken from <a href="https://ncatlab.org/nlab/show/simplicial+set#Friedman08">Friedman 08</a>)</p> <div class="num_defn" id="DegeneracyProjectionsInBarycentricCoords"> <h6 id="definition_3">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>k</mi><mo><</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0 \leq k \lt n</annotation></semantics></math> the <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>th degenerate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n)</annotation></semantics></math>-simplex (projection)</strong> is the surjective map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>k</mi></msub><mo>:</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>Δ</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> \sigma_k : \Delta^{n} \to \Delta^{n-1} </annotation></semantics></math></div> <p>induced under the barycentric coordinates of def. <a class="maruku-ref" href="#TopologicalSimplex"></a> under the surjection</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> \mathbb{R}^{n+1} \to \mathbb{R}^n </annotation></semantics></math></div> <p>which sends</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>k</mi></msub><mo>+</mo><msub><mi>x</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (x_0, \cdots, x_n) \mapsto (x_0, \cdots, x_{k} + x_{k+1}, \cdots, x_n) \,. </annotation></semantics></math></div></div> <div class="num_defn" id="SingularSimplex"> <h6 id="definition_4">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">X \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, a <strong>singular <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>-simplex</strong> in <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> is a <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous map</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>:</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \sigma : \Delta^n \to X </annotation></semantics></math></div> <p>from the topological <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>-simplex, def. <a class="maruku-ref" href="#TopologicalSimplex"></a>, to <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>.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Sing</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>≔</mo><msub><mi>Hom</mi> <mi>Top</mi></msub><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (Sing X)_n \coloneqq Hom_{Top}(\Delta^n , X) </annotation></semantics></math></div> <p>for the set of singular <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>-simplices of <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>.</p> </div> <p><img src="http://ncatlab.org/nlab/files/singularsimplices.jpg" width="500" /></p> <p>(graphics taken from <a href="https://ncatlab.org/nlab/show/simplicial+set#Friedman08">Friedman 08</a>)</p> <p>The sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Sing</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">(Sing X)_\bullet</annotation></semantics></math> here are closely related by an interlocking system of maps that make them form what is called a <em><a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a></em>, and as such the collection of these sets of singular simplices is called the <em><a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a></em> of <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>. We discuss the definition of simplicial sets now and then come back to this below in def. <a class="maruku-ref" href="#SingularSimplicialComplex"></a>.</p> <p>Since the topological <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>-simplices <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> from def. <a class="maruku-ref" href="#TopologicalSimplex"></a> sit inside each other by the face inclusions of def. <a class="maruku-ref" href="#FaceInclusionInBarycentricCoords"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>k</mi></msub><mo>:</mo><msup><mi>Δ</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> \delta_k : \Delta^{n-1} \to \Delta^{n} </annotation></semantics></math></div> <p>and project onto each other by the degeneracy maps, def. <a class="maruku-ref" href="#DegeneracyProjectionsInBarycentricCoords"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>k</mi></msub><mo>:</mo><msup><mi>Δ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> \sigma_k : \Delta^{n+1} \to \Delta^n </annotation></semantics></math></div> <p>we dually have functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>k</mi></msub><mo>≔</mo><msub><mi>Hom</mi> <mi>Top</mi></msub><mo stretchy="false">(</mo><msub><mi>δ</mi> <mi>k</mi></msub><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mi>Sing</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>→</mo><mo stretchy="false">(</mo><mi>Sing</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> d_k \coloneqq Hom_{Top}(\delta_k, X) : (Sing X)_n \to (Sing X)_{n-1} </annotation></semantics></math></div> <p>that send each singular <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>-simplex to its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-face and functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>k</mi></msub><mo>≔</mo><msub><mi>Hom</mi> <mi>Top</mi></msub><mo stretchy="false">(</mo><msub><mi>σ</mi> <mi>k</mi></msub><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mi>Sing</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>→</mo><mo stretchy="false">(</mo><mi>Sing</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> s_k \coloneqq Hom_{Top}(\sigma_k,X) : (Sing X)_{n} \to (Sing X)_{n+1} </annotation></semantics></math></div> <p>that regard an <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>-simplex as beign a degenerate (“thin”) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-simplex. All these sets of simplices and face and degeneracy maps between them form the following structure.</p> <div class="num_defn" id="SimplicialSet"> <h6 id="definition_5">Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">S \in sSet</annotation></semantics></math> is</p> <ul> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>n</mi></msub><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">S_n \in Set</annotation></semantics></math> – the <strong>set of <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>-<a class="existingWikiWord" href="/nlab/show/simplices">simplices</a></strong>;</p> </li> <li> <p>for each <a class="existingWikiWord" href="/nlab/show/injective+map">injective map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>i</mi></msub><mo>:</mo><mover><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>¯</mo></mover><mo>→</mo><mover><mi>n</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\delta_i : \overline{n-1} \to \overline{n}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/totally+ordered+sets">totally ordered sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>n</mi><mo stretchy="false">¯</mo></mover><mo>≔</mo><mo stretchy="false">{</mo><mn>0</mn><mo><</mo><mn>1</mn><mo><</mo><mi>⋯</mi><mo><</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\bar n \coloneqq \{ 0 \lt 1 \lt \cdots \lt n \}</annotation></semantics></math></p> <p>a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><mo>:</mo><msub><mi>S</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">d_i : S_{n} \to S_{n-1}</annotation></semantics></math> – the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>th <strong>face map</strong> 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>-simplices;</p> </li> <li> <p>for each <a class="existingWikiWord" href="/nlab/show/surjective+map">surjective map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>i</mi></msub><mo>:</mo><mover><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mo>¯</mo></mover><mo>→</mo><mover><mi>n</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\sigma_i : \overline{n+1} \to \bar n</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/totally+ordered+sets">totally ordered sets</a></p> <p>a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>i</mi></msub><mo>:</mo><msub><mi>S</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>S</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\sigma_i : S_{n} \to S_{n+1}</annotation></semantics></math> – the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>th <strong>degeneracy map</strong> 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>-simplices;</p> </li> </ul> <p>such that these functions satisfy the <em><a class="existingWikiWord" href="/nlab/show/simplicial+identities">simplicial identities</a></em>.</p> </div> <div class="num_defn" id="SimplicialIdentities"> <h6 id="definition_6">Definition</h6> <p>The <strong>simplicial identities</strong> satisfied by face and degeneracy maps as above are (whenever these maps are composable as indicated):</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>d</mi> <mi>j</mi></msub><mo>=</mo><msub><mi>d</mi> <mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>∘</mo><msub><mi>d</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex"> d_i \circ d_j = d_{j-1} \circ d_i</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo><</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">i \lt j</annotation></semantics></math>,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>s</mi> <mi>j</mi></msub><mo>=</mo><msub><mi>s</mi> <mi>j</mi></msub><mo>∘</mo><msub><mi>s</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">s_i \circ s_j = s_j \circ s_{i-1}</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>></mo><mi>j</mi></mrow><annotation encoding="application/x-tex">i \gt j</annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>s</mi> <mi>j</mi></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>s</mi> <mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>∘</mo><msub><mi>d</mi> <mi>i</mi></msub></mtd> <mtd><mi>if</mi><mspace width="thickmathspace"></mspace><mi>i</mi><mo><</mo><mi>j</mi></mtd></mtr> <mtr><mtd><mi>id</mi></mtd> <mtd><mi>if</mi><mspace width="thickmathspace"></mspace><mi>i</mi><mo>=</mo><mi>j</mi><mspace width="thickmathspace"></mspace><mi>or</mi><mspace width="thickmathspace"></mspace><mi>i</mi><mo>=</mo><mi>j</mi><mo>+</mo><mn>1</mn></mtd></mtr> <mtr><mtd><msub><mi>s</mi> <mi>j</mi></msub><mo>∘</mo><msub><mi>d</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mi>if</mi><mi>i</mi><mo>></mo><mi>j</mi><mo>+</mo><mn>1</mn></mtd></mtr></mtable></mrow></mrow></mrow><annotation encoding="application/x-tex">d_i \circ s_j = \left\{ \array{ s_{j-1} \circ d_i & if \; i \lt j \\ id & if \; i = j \; or \; i = j+1 \\ s_j \circ d_{i-1} & if i \gt j+1 } \right. </annotation></semantics></math></p> </li> </ol> </div> <p>It is straightforward to check by explicit inspection that the evident injection and restriction maps between the sets of <a class="existingWikiWord" href="/nlab/show/singular+simplices">singular simplices</a> make <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Sing</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">(Sing X)_\bullet</annotation></semantics></math> into a simplicial set. However for working with this, it is good to streamline a little:</p> <div class="num_defn"> <h6 id="definition_7">Definition</h6> <p>The <strong><a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> on the free categories of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mtd> <mtd><mo>≔</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mtd> <mtd><mo>≔</mo><mo stretchy="false">{</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mtd> <mtd><mo>≔</mo><mo stretchy="false">{</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo>→</mo><mn>2</mn><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} [0] & \coloneqq \{0\} \\ [1] & \coloneqq \{0 \to 1\} \\ [2] & \coloneqq \{0 \to 1 \to 2\} \\ \vdots \end{aligned} \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>This is called the “simplex category” because we are to think of the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[n]</annotation></semantics></math> as being the “<a class="existingWikiWord" href="/nlab/show/spine">spine</a>” of the <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>-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a>. For instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math> we think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mn>1</mn><mo>→</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">0 \to 1 \to 2</annotation></semantics></math> as the “spine” of the triangle. This becomes clear if we don’t just draw the morphisms that <em>generate</em> the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[n]</annotation></semantics></math>, but draw also all their composites. For instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math> we have_</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mn>2</mn></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [2] = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&\to&& 2 } \right\} \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> S : \Delta^{op} \to Set </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of the <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a> to the category <a class="existingWikiWord" href="/nlab/show/Set">Set</a> of sets is canonically identified with a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>, def. <a class="maruku-ref" href="#SimplicialSet"></a>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>One checks by inspection that the simplicial identities characterize precisely the behaviour of the morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>op</mi></msup><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta^{op}([n],[n+1])</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>op</mi></msup><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta^{op}([n],[n-1])</annotation></semantics></math>.</p> </div> <p>This makes the following evident:</p> <div class="num_example" id="StandardCosimplicialTopologicalSpace"> <h6 id="example_3">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/topological+simplices">topological simplices</a> from def. <a class="maruku-ref" href="#TopologicalSimplex"></a> arrange into a <em><a class="existingWikiWord" href="/nlab/show/cosimplicial+object">cosimplicial object</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a></em>, namely a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mo>•</mo></msup><mo>:</mo><mi>Δ</mi><mo>→</mo><mi>Top</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta^\bullet : \Delta \to Top \,. </annotation></semantics></math></div></div> <p>With this now the structure of a simplicial set on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Sing</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">(Sing X)_\bullet</annotation></semantics></math>, def. <a class="maruku-ref" href="#SingularSimplex"></a>, is manifest: it is just the <em><a class="existingWikiWord" href="/nlab/show/nerve">nerve</a></em> of <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> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">\Delta^\bullet</annotation></semantics></math>, namely:</p> <div class="num_defn" id="SingularSimplicialComplex"> <h6 id="definition_8">Definition</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> its <strong><a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">simplicial set of singular simplicies</a></strong> (often called the <strong><a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a></strong>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Sing</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> (Sing X)_\bullet : \Delta^{op} \to Set </annotation></semantics></math></div> <p>is given by composition of the functor from example <a class="maruku-ref" href="#StandardCosimplicialTopologicalSpace"></a> with the <a class="existingWikiWord" href="/nlab/show/hom+functor">hom functor</a> of <a class="existingWikiWord" href="/nlab/show/Top">Top</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Sing</mi><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↦</mo><msub><mi>Hom</mi> <mi>Top</mi></msub><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (Sing X) : [n] \mapsto Hom_{Top}( \Delta^n , X ) \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>It turns out – this is the content of the <em><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</em> (<a href="model+structure+on+simplicial+sets#Quillen67">Quillen 67</a>) – that <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of the topological space <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> is entirely captured by its singular simplicial complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">Sing X</annotation></semantics></math>. Moreover, the <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">Sing X</annotation></semantics></math> is a model for the same <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> as that of <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>, but with the special property that it is canonically a <a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a> – a <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a>. Better yet, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">Sing X</annotation></semantics></math> is itself already good cell complex, namely a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>. We come to this below.</p> </div> <h3 id="simplicial_homotopy">Simplicial homotopy</h3> <p>The concept of <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> of morphisms between simplicial sets proceeds in direct analogy with that in <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>.</p> <div class="num_defn" id="LeftHomotopyOfSimplicialSets"> <h6 id="definition_9">Definition</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>, def. <a class="maruku-ref" href="#SimplicialSet"></a>, its <em>simplicial <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></em> is the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">X\times \Delta[1]</annotation></semantics></math> (formed in the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>).</p> <p>A <em><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>f</mi><mo>⇒</mo><mi>g</mi></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; f \Rightarrow g </annotation></semantics></math></div> <p>between two morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> f,g\;\colon\; X \longrightarrow Y </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> is a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; X \times \Delta[1] \longrightarrow Y </annotation></semantics></math></div> <p>such that the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>d</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo>×</mo><msup><mi>Δ</mi> <mn>1</mn></msup></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>x</mi></msub><mo>,</mo><msub><mi>d</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0" lspace="-100%width"><mi>g</mi></mpadded></msub></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X \\ {}^{\mathllap{(id_X,d_1)}}\downarrow & \searrow^{\mathllap{f}} \\ X \times \Delta^1 &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{(id_x, d_0)}}\uparrow & \nearrow_{\mathllap{g}} \\ X } \,. </annotation></semantics></math></div> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>, def. <a class="maruku-ref" href="#SimplicialSet"></a>, its <em>simplicial <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a></em> is the <a class="existingWikiWord" href="/nlab/show/function+complex">function complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">X^{\Delta[1]}</annotation></semantics></math> (formed in the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>).</p> <p>A <em><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>f</mi><mo>⇒</mo><mi>g</mi></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; f \Rightarrow g </annotation></semantics></math></div> <p>between two morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> f,g\;\colon\; X \longrightarrow Y </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> is a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><msup><mi>Y</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> \eta \colon X \longrightarrow Y^{\Delta[1]} </annotation></semantics></math></div> <p>such that the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><msup><mi>Y</mi> <mrow><msub><mi>d</mi> <mn>1</mn></msub></mrow></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><msup><mi>Y</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>g</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>Y</mi> <mrow><msub><mi>d</mi> <mn>0</mn></msub></mrow></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && Y \\ & {}^{\mathllap{f}}\nearrow & \uparrow^{\mathrlap{Y^{d_1}}} \\ X &\stackrel{\eta}{\longrightarrow}& Y^{\Delta[1]} \\ & {}_{\mathllap{g}}\searrow & \downarrow^{\mathrlap{Y^{d_0}}} \\ && Y } \,. </annotation></semantics></math></div></div> <div class="num_prop" id="LeftHomotopyIsEquivalence"> <h6 id="proposition_2">Proposition</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>, def. <a class="maruku-ref" href="#KanComplexes"></a>, and <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> any <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>, then left homotopy, def. <a class="maruku-ref" href="#LeftHomotopyOfSimplicialSets"></a>, regarded as a <a class="existingWikiWord" href="/nlab/show/relation">relation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>∼</mo><mi>g</mi><mo stretchy="false">)</mo><mo>⇔</mo><mo stretchy="false">(</mo><mi>f</mi><mover><mo>⇒</mo><mo>∃</mo></mover><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (f\sim g) \Leftrightarrow (f \stackrel{\exists}{\Rightarrow} g) </annotation></semantics></math></div> <p>on the <a class="existingWikiWord" href="/nlab/show/hom+set">hom set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>sSet</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{sSet}(X,Y)</annotation></semantics></math>, is an <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a>.</p> </div> <div class="num_defn" id="HomotopyEquivalence"> <h6 id="definition_10">Definition</h6> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> is a left/right <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a> if there exists a morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⟵</mo><mi>Y</mi><mo lspace="verythinmathspace">:</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">X \longleftarrow Y \colon g</annotation></semantics></math> and left/right homotopies (def. <a class="maruku-ref" href="#LeftHomotopyOfSimplicialSets"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∘</mo><mi>f</mi><mo>⇒</mo><msub><mi>id</mi> <mi>X</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>f</mi><mo>∘</mo><mi>g</mi><mo>⇒</mo><msub><mi>id</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex"> g \circ f \Rightarrow id_X\,,\;\;\;\; f\circ g \Rightarrow id_Y </annotation></semantics></math></div></div> <p>The the basic invariants of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>/<a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a> in <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial homotopy theory</a> are their <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+groups">simplicial homotopy groups</a>, to which we turn now.</p> <p>Given that a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> is a special <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> that <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">behaves like</a> a combinatorial model for a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, the <em>simplicial homotopy groups</em> of a Kan complex are accordingly the combinatorial analog of the <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>: instead of being maps from topological <a class="existingWikiWord" href="/nlab/show/spheres">spheres</a> modulo maps from topological disks, they are maps from the <a class="existingWikiWord" href="/nlab/show/boundary+of+a+simplex">boundary of a simplex</a> modulo those from the <a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> itself.</p> <p>Accordingly, the definition of the discussion of simplicial homotopy groups is essentially literally the same as that of <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of topological spaces. One technical difference is for instance that the definition of the group structure is slightly more non-immediate for simplicial homotopy groups than for topological homotopy groups (see below).</p> <div class="num_defn" id="UnderlyingSetsOfSimplicialHomotopyGroups"> <h6 id="definition_11">Definition</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>, then its <strong>0th <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+group">simplicial homotopy group</a></strong> (or <strong>set of <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a></strong>) is the set of <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> of vertices modulo the <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>d</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>d</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mover><msub><mi>X</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_1 \stackrel{(d_1,d_0)}{\longrightarrow} X_0 \times X_0</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msub><mi>X</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>X</mi> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_0(X) \colon X_0/X_1 \,. </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x \in X_0</annotation></semantics></math> a vertex and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq 1</annotation></semantics></math>, then the underlying <a class="existingWikiWord" href="/nlab/show/set">set</a> of the <strong><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>th <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+group">simplicial homotopy group</a></strong> of <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> at <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> – denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(X,x)</annotation></semantics></math> – is, the set of <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\alpha]</annotation></semantics></math> of morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo lspace="verythinmathspace">:</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \alpha \colon \Delta^n \to X </annotation></semantics></math></div> <p>from the simplicial <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>-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> <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> to <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>, such that these take the <a class="existingWikiWord" href="/nlab/show/boundary+of+a+simplex">boundary of the simplex</a> to <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>, i.e. such that they fit into a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> of the form</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><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Δ</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>x</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>⟶</mo><mi>α</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \partial \Delta[n] & \longrightarrow & \Delta[0] \\ \downarrow && \downarrow^{\mathrlap{x}} \\ \Delta[n] &\stackrel{\alpha}{\longrightarrow}& X } \,, </annotation></semantics></math></div> <p>where two such maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>,</mo><mi>α</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\alpha, \alpha'</annotation></semantics></math> are taken to be equivalent is they are related by a <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy">simplicial homotopy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msup></mtd> <mtd><msup><mo>↘</mo> <mi>α</mi></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></msup></mtd> <mtd><msub><mo>↗</mo> <mrow><mi>α</mi><mo>′</mo></mrow></msub></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Delta[n] \\ \downarrow^{i_0} & \searrow^{\alpha} \\ \Delta[n] \times \Delta[1] &\stackrel{\eta}{\longrightarrow}& X \\ \uparrow^{i_1} & \nearrow_{\alpha'} \\ \Delta[n] } </annotation></semantics></math></div> <p>that fixes the boundary in that it fits into a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> of the form</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><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Δ</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>x</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \partial \Delta[n] \times \Delta[1] & \longrightarrow & \Delta[0] \\ \downarrow && \downarrow^{\mathrlap{x}} \\ \Delta[n] \times \Delta[1] &\stackrel{\eta}{\longrightarrow}& X } \,. </annotation></semantics></math></div></div> <p>These sets are taken to be equipped with the following group structure.</p> <div class="num_defn" id="ProductOnSimplicialHomotopyGroups"> <h6 id="definition_12">Definition</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x\in X_0</annotation></semantics></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq 1</annotation></semantics></math> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f,g \colon \Delta[n] \to X</annotation></semantics></math> two representatives of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(X,x)</annotation></semantics></math> as in def. <a class="maruku-ref" href="#UnderlyingSetsOfSimplicialHomotopyGroups"></a>, consider the following <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>-simplices in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_n</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub><mo>≔</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>s</mi> <mn>0</mn></msub><mo>∘</mo><msub><mi>s</mi> <mn>0</mn></msub><mo>∘</mo><mi>⋯</mi><mo>∘</mo><msub><mi>s</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mi>for</mi><mspace width="thickmathspace"></mspace><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>2</mn></mtd></mtr> <mtr><mtd><mi>f</mi></mtd> <mtd><mi>for</mi><mspace width="thickmathspace"></mspace><mi>i</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></mtd></mtr> <mtr><mtd><mi>g</mi></mtd> <mtd><mi>for</mi><mspace width="thickmathspace"></mspace><mi>i</mi><mo>=</mo><mi>n</mi><mo>+</mo><mn>1</mn></mtd></mtr></mtable></mrow></mrow></mrow><annotation encoding="application/x-tex"> v_i \coloneqq \left\{ \array{ s_0 \circ s_0 \circ \cdots \circ s_0 (x) & for \; 0 \leq i \leq n-2 \\ f & for \; i = n-1 \\ g & for \; i = n+1 } \right. </annotation></semantics></math></div> <p>This corresponds to a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Lambda^{n+1}[n] \to X</annotation></semantics></math> from a <a class="existingWikiWord" href="/nlab/show/horn">horn</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> into <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>. By the <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> property of <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> this morphism has an <a class="existingWikiWord" href="/nlab/show/extension">extension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> through the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[n]</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>Λ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mi>θ</mi></mpadded></msub></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Lambda^{n+1}[n] & \longrightarrow & X \\ \downarrow & \nearrow_{\mathrlap{\theta}} \\ \Delta[n+1] } </annotation></semantics></math></div> <p>From the <a class="existingWikiWord" href="/nlab/show/simplicial+identities">simplicial identities</a> one finds that the boundary of the <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>-simplex arising as the <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>th boundary piece <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>n</mi></msub><mi>θ</mi></mrow><annotation encoding="application/x-tex">d_n \theta</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> is constant on <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></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><msub><mi>d</mi> <mi>n</mi></msub><mi>θ</mi><mo>=</mo><msub><mi>d</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>d</mi> <mi>i</mi></msub><mi>θ</mi><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex"> d_i d_{n} \theta = d_{n-1} d_i \theta = x </annotation></semantics></math></div> <p>So <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>n</mi></msub><mi>θ</mi></mrow><annotation encoding="application/x-tex">d_n \theta</annotation></semantics></math> represents an element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(X,x)</annotation></semantics></math> and we define a product operation on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(X,x)</annotation></semantics></math> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo><mo>⋅</mo><mo stretchy="false">[</mo><mi>g</mi><mo stretchy="false">]</mo><mo>≔</mo><mo stretchy="false">[</mo><msub><mi>d</mi> <mi>n</mi></msub><mi>θ</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [f]\cdot [g] \coloneqq [d_n \theta] \,. </annotation></semantics></math></div></div> <p>(e.g. <a href="#GoerssJardine99">Goerss-Jardine 99, p. 26</a>)</p> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>All the degenerate <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>-simplices <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mrow><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">v_{0 \leq i \leq n-2}</annotation></semantics></math> in def. <a class="maruku-ref" href="#ProductOnSimplicialHomotopyGroups"></a> are just there so that the gluing of the two <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>-cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> to each other can be regarded as forming the boundary of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-simplex except for one face. By the Kan extension property that missing face exists, namely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>n</mi></msub><mi>θ</mi></mrow><annotation encoding="application/x-tex">d_n \theta</annotation></semantics></math>. This is a choice of gluing composite of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> with <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>.</p> </div> <div class="num_lemma"> <h6 id="lemma">Lemma</h6> <p>The product on homotopy group elements in def. <a class="maruku-ref" href="#ProductOnSimplicialHomotopyGroups"></a> is well defined, in that it is independent of the choice of representatives <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, <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> and of the extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math>.</p> </div> <p>e.g. (<a href="#GoerssJardine99">Goerss-Jardine 99, lemma 7.1</a>)</p> <div class="num_lemma"> <h6 id="lemma_2">Lemma</h6> <p>The product operation in def. <a class="maruku-ref" href="#ProductOnSimplicialHomotopyGroups"></a> yields a <a class="existingWikiWord" href="/nlab/show/group">group</a> structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(X,x)</annotation></semantics></math>, which is <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n \geq 2</annotation></semantics></math>.</p> </div> <p>e.g. (<a href="#GoerssJardine99">Goerss-Jardine 99, theorem 7.2</a>)</p> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>The first homotopy group, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X,x)</annotation></semantics></math>, is also called the <em><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></em> of <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>.</p> </div> <div class="num_defn" id="WeakHomotopyEquivalence"> <h6 id="definition_13">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>KanCplx</mi><mo>↪</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">X,Y \in KanCplx \hookrightarrow sSet</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a>, then a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> f \colon X \longrightarrow Y </annotation></semantics></math></div> <p>is called a <strong><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></strong> if it induces <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> on all <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+groups">simplicial homotopy groups</a>, i.e. if</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(f) \colon \pi_0(X) \longrightarrow \pi_0(Y)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> of sets;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(f,x) \colon \pi_n(X,x) \longrightarrow \pi_n(Y,f(x))</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of <a class="existingWikiWord" href="/nlab/show/groups">groups</a> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x\in X_0</annotation></semantics></math> and all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>; <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq 1</annotation></semantics></math>.</p> </li> </ol> </div> <h3 id="kan_fibrations">Kan fibrations</h3> <div class="num_defn" id="Horn"> <h6 id="definition_14">Definition</h6> <p>For each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0 \leq i \leq n</annotation></semantics></math>, the <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,i)</annotation></semantics></math>-horn</strong> is the subsimplicial set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↪</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \Lambda^i[n] \hookrightarrow \Delta[n] </annotation></semantics></math></div> <p>of the simplicial <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>-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a>, which is the <a class="existingWikiWord" href="/nlab/show/union">union</a> of all faces <em>except</em> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mi>th</mi></msup></mrow><annotation encoding="application/x-tex">i^{th}</annotation></semantics></math> one.</p> <p>This is called an <strong>outer horn</strong> if <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 = 0</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k = n</annotation></semantics></math>. Otherwise it is an <strong>inner horn</strong>.</p> </div> <p><img src="http://ncatlab.org/nlab/files/2horns.jpg" width="500" /></p> <p>(graphics taken from <a href="https://ncatlab.org/nlab/show/simplicial+set#Friedman08">Friedman 08</a>)</p> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>Since <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> is a <a class="existingWikiWord" href="/nlab/show/presheaf+category">presheaf category</a>, <a class="existingWikiWord" href="/nlab/show/unions">unions</a> of <a class="existingWikiWord" href="/nlab/show/subobjects">subobjects</a> make sense and they are calculated objectwise, thus in this case dimensionwise. This way it becomes clear what the structure of a horn as a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>k</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\Lambda^k[n]: \Delta^{op} \to Set</annotation></semantics></math> must therefore be: it takes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[m]</annotation></semantics></math> to the collection of ordinal maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f: [m] \to [n]</annotation></semantics></math> which do not have the element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> in the image.</p> </div> <div class="num_defn" id="KanComplexes"> <h6 id="definition_15">Definition</h6> <p>A <em><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></em> is a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> that satisfies the <em>Kan condition</em>,</p> <ul> <li> <p>which says that all <a class="existingWikiWord" href="/nlab/show/horns">horns</a> of the simplicial set have <em>fillers</em>/extend to <a class="existingWikiWord" href="/nlab/show/simplices">simplices</a>;</p> </li> <li> <p>which means equivalently that the unique homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>→</mo><mi>pt</mi></mrow><annotation encoding="application/x-tex">S \to pt</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/point">point</a> (the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal</a> <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>) is a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>;</p> </li> <li> <p>which means equivalently that for all <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a> of the form</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>Λ</mi> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>S</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>pt</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><msup><mi>Λ</mi> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>S</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Lambda^i[n] &\to& S \\ \downarrow && \downarrow \\ \Delta[n] &\to& pt } \;\;\; \leftrightarrow \;\;\; \array{ \Lambda^i[n] &\to& S \\ \downarrow && \\ \Delta[n] } </annotation></semantics></math></div> <p>there exists a diagonal morphism</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>Λ</mi> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>S</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>pt</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><msup><mi>Λ</mi> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>S</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Lambda^i[n] &\to& S \\ \downarrow &\nearrow& \downarrow \\ \Delta[n] &\to& pt } \;\;\; \leftrightarrow \;\;\; \array{ \Lambda^i[n] &\to& S \\ \downarrow &\nearrow& \\ \Delta[n] } </annotation></semantics></math></div> <p>completing this to a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a>;</p> </li> <li> <p>which in turn means equivalently that the map from <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>-simplices to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,i)</annotation></semantics></math>-horns is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mi>S</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>↠</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">[</mo><msup><mi>Λ</mi> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mi>S</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\Delta[n], S]\, \twoheadrightarrow \,[\Lambda^i[n],S] \,. </annotation></semantics></math></div></li> </ul> </div> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, its <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sing(X)</annotation></semantics></math>, def. <a class="maruku-ref" href="#SingularSimplicialComplex"></a>, is a Kan complex, def. <a class="maruku-ref" href="#KanComplexes"></a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>The inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow><msub><mrow><msup><mi>Λ</mi> <mi>n</mi></msup></mrow> <mi>Top</mi></msub></mrow> <mi>k</mi></msub><mo>↪</mo><msubsup><mi>Δ</mi> <mi>Top</mi> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">{{\Lambda^n}_{Top}}_k \hookrightarrow \Delta^n_{Top}</annotation></semantics></math> of topological horns into topological simplices are <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a>, in that there are <a class="existingWikiWord" href="/nlab/show/continuous+maps">continuous maps</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Δ</mi> <mi>Top</mi> <mi>n</mi></msubsup><mo>→</mo><msub><mrow><msub><mrow><msup><mi>Λ</mi> <mi>n</mi></msup></mrow> <mi>Top</mi></msub></mrow> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\Delta^n_{Top} \to {{\Lambda^n}_{Top}}_k</annotation></semantics></math> given by “squashing” a topological <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>-simplex onto parts of its boundary, such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mrow><msub><mrow><msup><mi>Λ</mi> <mi>n</mi></msup></mrow> <mi>Top</mi></msub></mrow> <mi>k</mi></msub><mo>→</mo><msubsup><mi>Δ</mi> <mi>Top</mi> <mi>n</mi></msubsup><mo>→</mo><msub><mrow><msub><mrow><msup><mi>Λ</mi> <mi>n</mi></msup></mrow> <mi>Top</mi></msub></mrow> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>Id</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ({{\Lambda^n}_{Top}}_k \to \Delta^n_{Top} \to {{\Lambda^n}_{Top}}_k) = Id \,. </annotation></semantics></math></div> <p>Therefore the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>,</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><msubsup><mi>Λ</mi> <mi>k</mi> <mi>n</mi></msubsup><mo>,</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Delta^n, \Pi(X)] \to [\Lambda^n_k,\Pi(X)]</annotation></semantics></math> is an epimorphism, since it is equal to to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Top</mi><mo stretchy="false">(</mo><msubsup><mi>Λ</mi> <mi>k</mi> <mi>n</mi></msubsup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Top(\Delta^n, X) \to Top(\Lambda^n_k, X)</annotation></semantics></math> which has a right inverse <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi><mo stretchy="false">(</mo><msubsup><mi>Λ</mi> <mi>k</mi> <mi>n</mi></msubsup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Top</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Top(\Lambda^n_k, X) \to Top(\Delta^n, X)</annotation></semantics></math>.</p> </div> <p>More generally:</p> <div class="num_defn" id="KanFibration"> <h6 id="definition_16">Definition</h6> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>S</mi><mo>⟶</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">\phi \colon S \longrightarrow T</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> is called a <em><a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a></em> if it has the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> again all <a class="existingWikiWord" href="/nlab/show/horn">horn</a> inclusions, def. <a class="maruku-ref" href="#Horn"></a>, hence if for every <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</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>Λ</mi> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>S</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ϕ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>T</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Lambda^i[n] &\longrightarrow& S \\ \downarrow && \downarrow^{\mathrlap{\phi}} \\ \Delta[n] &\longrightarrow& T } </annotation></semantics></math></div> <p>there exists a lift</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>Λ</mi> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>S</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ϕ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>T</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \Lambda^i[n] &\longrightarrow& S \\ \downarrow &\nearrow& \downarrow^{\mathrlap{\phi}} \\ \Delta[n] &\longrightarrow& T } \,. </annotation></semantics></math></div></div> <p>This is the simplicial incarnation of the concept of <a class="existingWikiWord" href="/nlab/show/Serre+fibrations">Serre fibrations</a> of topological spaces:</p> <div class="num_defn" id="SerreFibration"> <h6 id="definition_17">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</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 \longrightarrow Y</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> is a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a> if for all <a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and for every <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>C</mi><mo>×</mo><mi>I</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ C &\longrightarrow& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ C \times I &\longrightarrow& Y } </annotation></semantics></math></div> <p>there exists a lift</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>C</mi><mo>×</mo><mi>I</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ C &\longrightarrow& X \\ \downarrow &\nearrow& \downarrow^{\mathrlap{f}} \\ C \times I &\longrightarrow& Y } \,. </annotation></semantics></math></div></div> <div class="num_prop" id="SingDetextsAndReflectsFibrations"> <h6 id="proposition_4">Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</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 \longrightarrow Y</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>, def. <a class="maruku-ref" href="#SerreFibration"></a>, precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>Sing</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Sing</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sing(f) \colon Sing(X) \longrightarrow Sing(Y)</annotation></semantics></math> (def. <a class="maruku-ref" href="#SingularSimplicialComplex"></a>) is a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>, def. <a class="maruku-ref" href="#KanFibration"></a>.</p> </div> <p>The proof uses the basic tool of <a class="existingWikiWord" href="/nlab/show/nerve+and+realization">nerve and realization</a>-<a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> to which we get to below in prop. <a class="maruku-ref" href="#NerveAndRealizationAdjunction"></a>.</p> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>First observe that the left <a class="existingWikiWord" href="/nlab/show/lifting+property">lifting property</a> against all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>↪</mo><mi>C</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">C \hookrightarrow C \times I</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> is equivalent to left lifting against <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><msup><mi>Λ</mi> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">|</mo></mrow><mo>↪</mo><mrow><mo stretchy="false">|</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{\vert \Lambda^i[n]\vert} \hookrightarrow {\vert \Delta[n]\vert}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/horn">horn</a> inclusions. Then apply the <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>sSet</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>Sing</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Top({\vert-\vert},-) \simeq sSet(-,Sing(-))</annotation></semantics></math>, given by the <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> of prop. <a class="maruku-ref" href="#NerveAndRealizationAdjunction"></a> and example <a class="maruku-ref" href="#TopologicalRealizationOfSimplicialSets"></a>, to the lifting diagrams.</p> </div> <div class="num_lemma" id="PullbackOfKanFibrationSendsLeftHomotopyToFiberwiseHomotopyequivalence"> <h6 id="lemma_3">Lemma</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p \colon X \longrightarrow Y</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>, def. <a class="maruku-ref" href="#KanFibration"></a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f_1,f_2 \colon A \longrightarrow X</annotation></semantics></math> be two morphisms. If there is a <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a> (def. <a class="maruku-ref" href="#LeftHomotopyOfSimplicialSets"></a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub><mo>⇒</mo><msub><mi>f</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f_1 \Rightarrow f_2</annotation></semantics></math> between these maps, then there is a fiberwise <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, def. <a class="maruku-ref" href="#HomotopyEquivalence"></a>, between the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> fibrations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>f</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>X</mi><mo>≃</mo><msubsup><mi>f</mi> <mn>2</mn> <mo>*</mo></msubsup><mi>X</mi></mrow><annotation encoding="application/x-tex">f_1^\ast X \simeq f_2^\ast X</annotation></semantics></math>.</p> </div> <p>(e.g. <a href="#GoerssJardine99">Goerss-Jardine 99, chapter I, lemma 10.6</a>)</p> <p>While <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> have the advantage of being purely combinatorial structures, the <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> of any given <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, def. <a class="maruku-ref" href="#SingularSimplicialComplex"></a> is in general a huge simplicial set which does not lend itself to detailed inspection. The following is about small models.</p> <div class="num_defn" id="MinimalKanFibration"> <h6 id="definition_18">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>S</mi><mo>⟶</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">\phi \colon S \longrightarrow T</annotation></semantics></math>, def. <a class="maruku-ref" href="#KanFibration"></a>, is called a <strong><a class="existingWikiWord" href="/nlab/show/minimal+Kan+fibration">minimal Kan fibration</a></strong> if for any two cells in the same fiber with the same <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> if they are homotopic relative their boundary, then they are already equal.</p> <p>More formally, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> is minimal precisely if for every <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</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 stretchy="false">(</mo><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>⟶</mo><mi>h</mi></mover></mtd> <mtd><mi>S</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ϕ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>T</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (\partial \Delta[n]) \times \Delta[1] &\stackrel{p_1}{\longrightarrow}& \partial \Delta[n] \\ \downarrow && \downarrow \\ \Delta[n] \times \Delta[1] &\stackrel{h}{\longrightarrow}& S \\ \downarrow^{\mathrlap{p_1}} && \downarrow^{\mathrlap{\phi}} \\ \Delta[n] &\longrightarrow& T } </annotation></semantics></math></div> <p>then the two composites</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mover><munder><mo>⟶</mo><mrow><msub><mi>d</mi> <mn>1</mn></msub></mrow></munder><mover><mo>⟶</mo><mrow><msub><mi>d</mi> <mn>0</mn></msub></mrow></mover></mover><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mover><mo>⟶</mo><mi>h</mi></mover><mi>S</mi></mrow><annotation encoding="application/x-tex"> \Delta[n] \stackrel{\overset{d_0}{\longrightarrow}}{\underset{d_1}{\longrightarrow}} \Delta[n] \times \Delta[1] \stackrel{h}{\longrightarrow} S </annotation></semantics></math></div> <p>are equal.</p> </div> <div class="num_prop" id="PullbackPreservesMinimalFibration"> <h6 id="proposition_5">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> (in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>) of a <a class="existingWikiWord" href="/nlab/show/minimal+Kan+fibration">minimal Kan fibration</a>, def. <a class="maruku-ref" href="#MinimalKanFibration"></a>, along any morphism is again a mimimal Kan fibration.</p> </div> <p>… <a class="existingWikiWord" href="/nlab/show/anodyne+extensions">anodyne extensions</a>…</p> <p>(<a href="#GoerssJardine99">Goerss-Jardine 99, chapter I, section 4</a>, <a href="#JoyalTierney09">Joyal-Tierney 09, section 1.7</a>)</p> <div class="num_prop" id="KanFibrationHasMinimalStrongDeformationRetract"> <h6 id="proposition_6">Proposition</h6> <p>For every <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>, def. <a class="maruku-ref" href="#KanFibration"></a>, there exists a fiberwise <a class="existingWikiWord" href="/nlab/show/strong+deformation+retract">strong deformation retract</a> to a <a class="existingWikiWord" href="/nlab/show/minimal+Kan+fibration">minimal Kan fibration</a>, def. <a class="maruku-ref" href="#MinimalKanFibration"></a>.</p> </div> <p>(e.g. <a href="#GoerssJardine99">Goerss-Jardine 99, chapter I, prop. 10.3</a>, <a href="#JoyalTierney08">Joyal-Tierney 08, theorem 3.3.1, theorem 3.3.3</a>).</p> <div class="proof"> <h6 id="proof_idea">Proof idea</h6> <p>Choose representatives by <a class="existingWikiWord" href="/nlab/show/induction">induction</a>, use that in the induction step one needs lifts of <a class="existingWikiWord" href="/nlab/show/anodyne+extensions">anodyne extensions</a> against a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>, which exist.</p> </div> <div class="num_lemma" id="FiberwiseHomotopyEquivalenceOfMinimalFibrationsIsIso"> <h6 id="lemma_4">Lemma</h6> <p>A morphism between <a class="existingWikiWord" href="/nlab/show/minimal+Kan+fibrations">minimal Kan fibrations</a>, def. <a class="maruku-ref" href="#MinimalKanFibration"></a>, which is fiberwise a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, def. <a class="maruku-ref" href="#HomotopyEquivalence"></a>, is already an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> </div> <p>(e.g. <a href="#GoerssJardine99">Goerss-Jardine 99, chapter I, lemma 10.4</a>)</p> <div class="proof"> <h6 id="proof_idea_2">Proof idea</h6> <p>Show the statement degreewise. In the <a class="existingWikiWord" href="/nlab/show/induction">induction</a> one needs to lift <a class="existingWikiWord" href="/nlab/show/anodyne+extensions">anodyne extensions</a> agains a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>.</p> </div> <div class="num_lemma" id="MinimalKanFibrationAreFiberBundles"> <h6 id="lemma_5">Lemma</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/minimal+Kan+fibration">minimal Kan fibration</a>, def. <a class="maruku-ref" href="#MinimalKanFibration"></a>, over a <a class="existingWikiWord" href="/nlab/show/connected">connected</a> base is a simplicial <a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a>, locally trivial over every simplex of the base.</p> </div> <p>(e.g. <a href="#GoerssJardine99">Goerss-Jardine 99, chapter I, corollary 10.8</a>)</p> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>By assumption of the base being connected, the classifying maps for the fibers over any two vertices are connected by a <a class="existingWikiWord" href="/nlab/show/zig-zag">zig-zag</a> of <a class="existingWikiWord" href="/nlab/show/homotopies">homotopies</a>, hence by lemma <a class="maruku-ref" href="#PullbackOfKanFibrationSendsLeftHomotopyToFiberwiseHomotopyequivalence"></a> the fibers are connected by <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a> and then by prop. <a class="maruku-ref" href="#PullbackPreservesMinimalFibration"></a> and lemma <a class="maruku-ref" href="#FiberwiseHomotopyEquivalenceOfMinimalFibrationsIsIso"></a> they are already isomorphic. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> for this <a class="existingWikiWord" href="/nlab/show/typical+fiber">typical fiber</a>.</p> <p>Moreover, for all <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> the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo><mo>→</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[n] \to \Delta[0] \to \Delta[n]</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopic</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mover><mo>→</mo><mi>id</mi></mover><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[n] \stackrel{id}{\to} \Delta[n]</annotation></semantics></math> and so applying lemma <a class="maruku-ref" href="#PullbackOfKanFibrationSendsLeftHomotopyToFiberwiseHomotopyequivalence"></a> and prop. <a class="maruku-ref" href="#FiberwiseHomotopyEquivalenceOfMinimalFibrationsIsIso"></a> once more yields that the fiber over each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[n]</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>×</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\Delta[n]\times F</annotation></semantics></math>.</p> </div> <h3 id="geometric_realization">Geometric realization</h3> <p>So far we we have considered passing from <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> to <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> by applying the <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> functor of def. <a class="maruku-ref" href="#SingularSimplicialComplex"></a>. Now we discuss a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> of this functor, called <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a>, which turns a simplicial set into a topological space by identifying each of its abstract <a class="existingWikiWord" href="/nlab/show/n-simplices">n-simplices</a> with the standard topological <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>-simplex.</p> <p>This is an example of a general abstract phenomenon:</p> <div class="num_prop" id="NerveAndRealizationAdjunction"> <h6 id="proposition_7">Proposition</h6> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>D</mi><mo>⟶</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> \delta \;\colon\; D \longrightarrow \mathcal{C} </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> from a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> <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> to a <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> with all <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>. Then the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a>-functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo>⟶</mo><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> N \;\colon\; \mathcal{C} \longrightarrow [D^{op}, Set] </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> N(X) \coloneqq \mathcal{C}(\delta(-),X) </annotation></semantics></math></div> <p>has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{\vert-\vert}</annotation></semantics></math>, called <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mo>⊣</mo><mi>N</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mover><munder><mo>⟶</mo><mi>N</mi></munder><mover><mo>⟵</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mover></mover><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> ({\vert-\vert} \dashv N) \;\colon\; \mathcal{C} \stackrel{\overset{{\vert-\vert}}{\longleftarrow}}{\underset{N}{\longrightarrow}} [D^{op}, Set] </annotation></semantics></math></div> <p>given by the <a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>S</mi><mo stretchy="false">|</mo></mrow><mo>=</mo><msup><mo>∫</mo> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></msup><mi>δ</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>S</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> {\vert S\vert} = \int^{d \in D} \delta(d) \cdot S(d) \,. </annotation></semantics></math></div></div> <p>(<a href="nerve+and+realization#Kan58">Kan 58</a>)</p> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>By basic propeties of <a class="existingWikiWord" href="/nlab/show/ends">ends</a> and <a class="existingWikiWord" href="/nlab/show/coends">coends</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>N</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><msub><mo>∫</mo> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></msub><mi>Set</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mi>N</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mo>∫</mo> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></msub><mi>Set</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>δ</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mo>∫</mo> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></msub><mi>𝒞</mi><mo stretchy="false">(</mo><mi>δ</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>S</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>𝒞</mi><mo stretchy="false">(</mo><msup><mo>∫</mo> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></msup><mi>δ</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>S</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mi>S</mi><mo stretchy="false">|</mo></mrow><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} [D^{op}, Set](S,N(X)) & = \int_{d \in D} Set(S(d), N(X)(d)) \\ & = \int_{d\in D} Set(S(d), \mathcal{C}(\delta(d),X)) \\ & \simeq \int_{d \in D} \mathcal{C}(\delta(d) \cdot S(d), X) \\ & \simeq \mathcal{C}(\int^{d \in D} \delta(d) \cdot S(d), X) \\ & = \mathcal{C}({\vert S\vert}, X) \,. \end{aligned} </annotation></semantics></math></div></div> <div class="num_example" id="TopologicalRealizationOfSimplicialSets"> <h6 id="example_4">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi></mrow><annotation encoding="application/x-tex">Sing</annotation></semantics></math> of def. <a class="maruku-ref" href="#SingularSimplicialComplex"></a> has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mo lspace="verythinmathspace">:</mo><mi>sSet</mi><mo>⟶</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> {\vert-\vert} \colon sSet \longrightarrow Top </annotation></semantics></math></div> <p>given by the <a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>S</mi><mo stretchy="false">|</mo></mrow><mo>=</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><msup><mi>Δ</mi> <mi>n</mi></msup><mo>⋅</mo><msub><mi>S</mi> <mi>n</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> {\vert S \vert} = \int^{[n]\in \Delta} \Delta^n \cdot S_n \,. </annotation></semantics></math></div></div> <p>Topological geometric realization takes values in particularly nice topological spaces.</p> <div class="num_defn" id="TopologicalRealizationOfsSetLandsInCWComplexes"> <h6 id="proposition_8">Proposition</h6> <p>The topological <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> in example <a class="maruku-ref" href="#TopologicalRealizationOfSimplicialSets"></a> takes values in <a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a>.</p> </div> <p>(e.g. <a href="#GoerssJardine99">Goerss-Jardine 99, chapter I, prop. 2.3</a>)</p> <p>Thus for a topological space <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> the <a class="existingWikiWord" href="/nlab/show/adjunction+counit">adjunction counit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϵ</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><mrow><mo stretchy="false">|</mo><mi>Sing</mi><mi>X</mi><mo stretchy="false">|</mo></mrow><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\epsilon_X \colon {\vert Sing X\vert} \longrightarrow X</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/nerve+and+realization">nerve and realization</a>-adjunction is a candidate for a replacement of <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> by a CW-complex. For this, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϵ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\epsilon_X</annotation></semantics></math> should be at least a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>, i.e. induce <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> on all <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a>. Since homotopy groups are built from maps into <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> out of <a class="existingWikiWord" href="/nlab/show/compact+topological+spaces">compact topological spaces</a> it is plausible that this works if the topology of <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> is entirely detected by maps out of compact topological spaces into <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>. Topological spaces with this property are called <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated</a>.</p> <p>We take <em><a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact topological space</a></em> to imply <em><a class="existingWikiWord" href="/nlab/show/Hausdorff+topological+space">Hausdorff topological space</a></em>.</p> <div class="num_defn" id="kTop"> <h6 id="definition_19">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \subset X</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <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> is called <strong>compactly open</strong> or <strong>compactly closed</strong>, respectively, if for every <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</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>K</mi><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f \colon K \longrightarrow X</annotation></semantics></math> out of a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact topological space</a> the <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">f^{-1}(U) \subset K</annotation></semantics></math> is open or closed, respectively.</p> <p>A topological space <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> is a <strong><a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+space">compactly generated topological space</a></strong> if each of its compactly closed subspaces is already closed.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub><mo>↪</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> Top_{cg} \hookrightarrow Top </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <a class="existingWikiWord" href="/nlab/show/Top">Top</a> on the compactly generated topological spaces.</p> </div> <p>Often the condition is added that a compactly closed topological space be also a <a class="existingWikiWord" href="/nlab/show/weakly+Hausdorff+topological+space">weakly Hausdorff topological space</a>.</p> <div class="num_example" id="ExamplesOfCompactlyGeneratedTopologiclSpaces"> <h6 id="example_5">Example</h6> <p>Examples of <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a>, def. <a class="maruku-ref" href="#kTop"></a>, include</p> <ul> <li> <p>every <a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>;</p> </li> <li> <p>every <a class="existingWikiWord" href="/nlab/show/locally+compact+space">locally compact space</a>;</p> </li> <li> <p>every <a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a>;</p> </li> <li> <p>every <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a>;</p> </li> <li> <p>every <a class="existingWikiWord" href="/nlab/show/first+countable+space">first countable space</a></p> </li> </ul> </div> <div class="num_cor" id="TopologicalRealizationOfSSetLandsInkTop"> <h6 id="corollary">Corollary</h6> <p>The topological <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> functor of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> in example <a class="maruku-ref" href="#TopologicalRealizationOfSimplicialSets"></a> takes values in <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>sSet</mi><mo>⟶</mo><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex"> {\vert - \vert} \;\colon\; sSet \longrightarrow Top_{cg} </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>By example <a class="maruku-ref" href="#ExamplesOfCompactlyGeneratedTopologiclSpaces"></a> and prop. <a class="maruku-ref" href="#TopologicalRealizationOfsSetLandsInCWComplexes"></a>.</p> </div> <div class="num_prop" id="kTopIsCoreflectiveInTop"> <h6 id="proposition_9">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub><mo>↪</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top_{cg} \hookrightarrow Top</annotation></semantics></math> of def. <a class="maruku-ref" href="#kTop"></a> has the following properties</p> <ol> <li> <p>It is a <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflective subcategory</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub><mover><munder><mo>⟵</mo><mi>k</mi></munder><mo>↪</mo></mover><mi>Top</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Top_{cg} \stackrel{\hookrightarrow}{\underset{k}{\longleftarrow}} Top \,. </annotation></semantics></math></div> <p>The coreflection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k(X)</annotation></semantics></math> of a topological space is given by adding to the open subsets of <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> all compactly open subsets, def. <a class="maruku-ref" href="#kTop"></a>.</p> </li> <li> <p>It has all small <a class="existingWikiWord" href="/nlab/show/limits">limits</a> and <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>.</p> <p>The colimits are computed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>, the limits are the image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> of the limits as computed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>.</p> </li> <li> <p>It is a <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math> is the image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> of the Cartesian product formed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>.</p> </li> </ol> </div> <p>This is due to (<a href="compactly+generated+topological+space#Steenrod67">Steenrod 67</a>), expanded on in (<a href="compactly+generated+topological+space#Lewis78">Lewis 78, appendix A</a>). One says that prop. <a class="maruku-ref" href="#kTopIsCoreflectiveInTop"></a> with example <a class="maruku-ref" href="#ExamplesOfCompactlyGeneratedTopologiclSpaces"></a> makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math> a “<a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a>”.</p> <div class="num_prop" id="Timesk"> <h6 id="proposition_10">Proposition</h6> <p>Regarded, via corollary <a class="maruku-ref" href="#TopologicalRealizationOfSSetLandsInkTop"></a> as a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mo lspace="verythinmathspace">:</mo><mi>sSet</mi><mo>→</mo><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">{\vert - \vert} \colon sSet \to Top_{cg}</annotation></semantics></math>, <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> preserves <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a>.</p> </div> <p>See at <em><a href="geometric+realization#GeometricRealizationIsLeftExact">Geometric realization is left exact</a></em>.</p> <div class="proof"> <h6 id="proof_idea_3">Proof idea</h6> <p>The key step in the proof is to use the <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closure</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math> (prop. <a class="maruku-ref" href="#kTopIsCoreflectiveInTop"></a>). This gives that the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> is a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> and hence preserves colimits in each variable, so that the <a class="existingWikiWord" href="/nlab/show/coend">coend</a> in the definition of the geometric realization may be taken out of Cartesian products.</p> </div> <div class="num_lemma"> <h6 id="lemma_6">Lemma</h6> <p>The <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a>, example <a class="maruku-ref" href="#TopologicalRealizationOfSimplicialSets"></a>, of a <a class="existingWikiWord" href="/nlab/show/minimal+Kan+fibration">minimal Kan fibration</a>, def. <a class="maruku-ref" href="#MinimalKanFibration"></a> is a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>, def. <a class="maruku-ref" href="#SerreFibration"></a>.</p> </div> <p>This is due to (<a class="existingWikiWord" href="/nlab/show/Calculus+of+fractions+and+homotopy+theory">Gabriel-Zisman 67</a>). See for instance (<a href="#GoerssJardine99">Goerss-Jardine 99, chapter I, corollary 10.8, theorem 10.9</a>).</p> <div class="proof"> <h6 id="proof_idea_4">Proof idea</h6> <p>By prop. <a class="maruku-ref" href="#MinimalKanFibrationAreFiberBundles"></a> minimal Kan fibrations are simplicial <a class="existingWikiWord" href="/nlab/show/fiber+bundles">fiber bundles</a>, locally trivial over each simplex in the base. By prop. <a class="maruku-ref" href="#Timesk"></a> this property translates to their <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> also being a locally trivial <a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, hence in particular a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>.</p> </div> <div class="num_prop" id="GeometricRealizationOfKanFibrationIsSerreFibration"> <h6 id="proposition_11">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a>, example <a class="maruku-ref" href="#TopologicalRealizationOfSimplicialSets"></a>, of any <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>, def. <a class="maruku-ref" href="#KanFibration"></a> is a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>, def. <a class="maruku-ref" href="#SerreFibration"></a>.</p> </div> <p>This is due to (<a href="Kan+fibration#Quillen68">Quillen 68</a>). See for instance (<a href="#GoerssJardine99">Goerss-Jardine 99, chapter I, theorem 10.10</a>).</p> <div class="num_prop" id="UnitOfSingularNerveAndRealizationIsWEOnKanComplexes"> <h6 id="proposition_12">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>, then the <a class="existingWikiWord" href="/nlab/show/adjunction+unit">unit</a> of the <a class="existingWikiWord" href="/nlab/show/nerve+and+realization">nerve and realization</a>-<a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> (prop. <a class="maruku-ref" href="#NerveAndRealizationAdjunction"></a>, example <a class="maruku-ref" href="#TopologicalRealizationOfSimplicialSets"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⟶</mo><mi>Sing</mi><mrow><mo stretchy="false">|</mo><mi>S</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex"> S \longrightarrow Sing {\vert S \vert} </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>, def. <a class="maruku-ref" href="#WeakHomotopyEquivalence"></a>.</p> <p>For <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> any <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, then the <a class="existingWikiWord" href="/nlab/show/adjunction+counit">adjunction counit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>Sing</mi><mi>X</mi><mo stretchy="false">|</mo></mrow><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> {\vert Sing X\vert} \longrightarrow X </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> </div> <p>e.g. (<a href="#GoerssJardine99">Goerss-Jardine 99, chapter I, prop. 11.1 and p. 63</a>).</p> <div class="proof"> <h6 id="proof_idea_5">Proof idea</h6> <p>Use prop. <a class="maruku-ref" href="#SingDetextsAndReflectsFibrations"></a> and prop. <a class="maruku-ref" href="#GeometricRealizationOfKanFibrationIsSerreFibration"></a> applied to the <a class="existingWikiWord" href="/nlab/show/path+fibration">path fibration</a> to proceed by <a class="existingWikiWord" href="/nlab/show/induction">induction</a>.</p> </div> <h2 id="TheClassicalModelStructure">The classical model structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math></h2> <div class="num_defn" id="ClassesOfMorphismsOnsSetQuillen"> <h6 id="definition_20">Definition</h6> <p>The classical model structure on <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math>, has the following distinguished classes of morphisms:</p> <ul> <li> <p>The classical <strong>weak equivalences</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/simplicial+weak+equivalences">simplicial weak equivalences</a>: morphisms whose <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a>, example <a class="maruku-ref" href="#TopologicalRealizationOfSimplicialSets"></a>, is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>;</p> </li> <li> <p>The classical <strong>fibrations</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> are the <strong><a class="existingWikiWord" href="/nlab/show/Kan+fibrations">Kan fibrations</a></strong>, def. <a class="maruku-ref" href="#KanFibration"></a>;</p> </li> <li> <p>The classical <strong>cofibrations</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> of simplicial sets, i.e. the degreewise <a class="existingWikiWord" href="/nlab/show/injections">injections</a>.</p> </li> </ul> </div> <h2 id="properties">Properties</h2> <h3 id="basic_properties">Basic properties</h3> <div class="num_prop"> <h6 id="proposition_13">Proposition</h6> <p>In model structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math>, def. <a class="maruku-ref" href="#ClassesOfMorphismsOnsSetQuillen"></a>, the following holds.</p> <ul> <li> <p>The fibrant objects are precisely the <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a>.</p> </li> <li> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> of fibrant simplicial sets / <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a> is a weak equivalence precisely if it induces an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> on all <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+groups">simplicial homotopy groups</a>, def. <a class="maruku-ref" href="#UnderlyingSetsOfSimplicialHomotopyGroups"></a>.</p> </li> <li> <p>All simplicial sets are cofibrant with respect to this model structure.</p> </li> </ul> </div> <div class="num_prop" id="AcyclicKanFibrationsAsRLPAgainstBoundaryInclusions"> <h6 id="proposition_14">Proposition</h6> <p>The <strong><a class="existingWikiWord" href="/nlab/show/acyclic+fibrations">acyclic fibrations</a></strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math>, namely the <em><a class="existingWikiWord" href="/nlab/show/acyclic+Kan+fibrations">acyclic Kan fibrations</a></em> (i.e. the maps that are both <a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a> as well as <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a>) are precisely the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \,\colon\, X \to Y</annotation></semantics></math> that have the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> with respect to all inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↪</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\partial \Delta[n] \hookrightarrow \Delta[n]</annotation></semantics></math> of boundaries of <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>-simplices into their <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>-simplices</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><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mo>∃</mo></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>f</mi></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \partial \Delta[n] &\to& X \\ \downarrow &{}^\exists\nearrow& \downarrow^f \\ \Delta[n] &\to& Y } \,. </annotation></semantics></math></div></div> <p>This appears spelled out for instance as (<a href="#GoerssJardine99">Goerss-Jardine 99, theorem 11.2</a>).</p> <p>In fact:</p> <div class="num_prop" id="AsACofibrantlyGeneratedModelCategory"> <h6 id="proposition_15">Proposition</h6> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a> with</p> <ul> <li> <p>generating cofibrations the <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\partial \Delta[n] \to \Delta[n]</annotation></semantics></math>;</p> </li> <li> <p>generating acyclic cofibrations the <a class="existingWikiWord" href="/nlab/show/horn">horn</a> inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Lambda^i[n] \to \Delta[n]</annotation></semantics></math>.</p> </li> </ul> </div> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> be the smallest class of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math> satisfying the following conditions:</p> <ol> <li>The class of monomorphisms that are in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> is closed under <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>, <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a>, and <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a>.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> has the <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> property in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math> and contains all the <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>.</li> <li>For all natural numbers <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>, the unique morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta [n] \to \Delta [0]</annotation></semantics></math> is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</li> </ol> <p>Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> is the class of weak homotopy equivalences.</p> </div> <div class="proof"> <h6 id="proof_7">Proof</h6> <ul> <li>First, notice that the horn inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mn>0</mn></msup><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>↪</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Lambda^0 [1] \hookrightarrow \Delta [1]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mn>1</mn></msup><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>↪</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Lambda^1 [1] \hookrightarrow \Delta [1]</annotation></semantics></math> are in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</li> <li>Suppose that the horn inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>k</mi></msup><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo><mo>↪</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Lambda^k [m] \hookrightarrow \Delta [m]</annotation></semantics></math> is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo><</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m \lt n</annotation></semantics></math> and all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">0 \le k \le m</annotation></semantics></math>. Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>l</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0 \le l \le n</annotation></semantics></math>, the horn inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>l</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↪</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Lambda^l [n] \hookrightarrow \Delta [n]</annotation></semantics></math> is also in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</li> <li>Quillen’s <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a> then implies all the trivial cofibrations are in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p : X \to Y</annotation></semantics></math> is a trivial Kan fibration, then its right lifting property implies there is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">s : Y \to X</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∘</mo><mi>s</mi><mo>=</mo><msub><mi>id</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex">p \circ s = id_Y</annotation></semantics></math>, and the two-out-of-three property implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">s : Y \to X</annotation></semantics></math> is a trivial cofibration. Thus every trivial Kan fibration is also in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</li> <li>Every weak homotopy equivalence factors as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∘</mo><mi>i</mi></mrow><annotation encoding="application/x-tex">p \circ i</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a trivial Kan fibration and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is a trivial cofibration, so every weak homotopy equivalence is indeed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</li> <li>Finally, noting that the class of weak homotopy equivalences satisfies the conditions in the theorem, we deduce that it is the <em>smallest</em> such class.</li> </ul> </div> <p>As a corollary, we deduce that the classical model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math> is the smallest (in terms of weak equivalences) model structure for which the cofibrations are the monomorphisms and the weak equivalences include the (combinatorial) homotopy equivalences.</p> <div class="num_prop"> <h6 id="proposition_16">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo>:</mo><mi>sSet</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\pi_0 : sSet \to Set</annotation></semantics></math> be the connected components functor, i.e. the left adjoint of the constant functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>cst</mi><mo>:</mo><mi>Set</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">cst : Set \to sSet</annotation></semantics></math>. A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Z</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">f : Z \to W</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math> is a weak homotopy equivalence if and only if the induced map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><msup><mi>K</mi> <mi>f</mi></msup><mo>:</mo><msub><mi>π</mi> <mn>0</mn></msub><msup><mi>K</mi> <mi>W</mi></msup><mo>→</mo><msub><mi>π</mi> <mn>0</mn></msub><msup><mi>K</mi> <mi>Z</mi></msup></mrow><annotation encoding="application/x-tex">\pi_0 K^f : \pi_0 K^W \to \pi_0 K^Z</annotation></semantics></math></div> <p>is a bijection for all <em>Kan complexes</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_8">Proof</h6> <p>One direction is easy: if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is a Kan complex, then axiom SM7 for <a class="existingWikiWord" href="/nlab/show/simplicial+model+categories">simplicial model categories</a> implies the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msup><mo>:</mo><msup><mi>sSet</mi> <mi>op</mi></msup><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">K^{(-)} : sSet^{op} \to sSet</annotation></semantics></math> is a right <a class="existingWikiWord" href="/nlab/show/Quillen+functor">Quillen functor</a>, so Ken Brown’s lemma implies it preserves all weak homotopy equivalences; in particular, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><msup><mi>K</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msup><mo>:</mo><msup><mi>sSet</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\pi_0 K^{(-)} : sSet^{op} \to Set</annotation></semantics></math> sends weak homotopy equivalences to bijections.</p> <p>Conversely, when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is a Kan complex, there is a natural bijection between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><msup><mi>K</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">\pi_0 K^X</annotation></semantics></math> and the hom-set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>sSet</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho (sSet) (X, K)</annotation></semantics></math>, and thus by the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>, a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Z</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">f : Z \to W</annotation></semantics></math> such that the induced morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><msup><mi>K</mi> <mi>W</mi></msup><mo>→</mo><msub><mi>π</mi> <mn>0</mn></msub><msup><mi>K</mi> <mi>Z</mi></msup></mrow><annotation encoding="application/x-tex">\pi_0 K^W \to \pi_0 K^Z</annotation></semantics></math> is a bijection for all Kan complexes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is precisely a morphism that becomes an isomorphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>sSet</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho (sSet)</annotation></semantics></math>, i.e. a weak homotopy equivalence.</p> </div> <h3 id="Properness">Properness</h3> <p>The Quillen model structure is both left and right <a class="existingWikiWord" href="/nlab/show/proper+model+category">proper</a>. Left properness is automatic since all objects are cofibrant. Right properness follows from the following argument: it suffices to show that there is a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> which (1) preserves fibrations, (2) preserves pullbacks of fibrations, (3) preserves and reflects weak equivalences, and (4) lands in a category in which the pullback of a weak equivalence along a fibration is a weak equivalence. For if so, we can apply <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> to the pullback of a fibration along a weak equivalence to get another such pullback in the codomain of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, which is a weak equivalence, and hence the original pullback was also a weak equivalence. Two such functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> are</p> <ul> <li>geometric realization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">sSet \to Top</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> denotes a sufficiently <a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a> (e.g. the category of <a class="existingWikiWord" href="/nlab/show/k-spaces">k-spaces</a> suffices) and</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ex</mi> <mn>∞</mn></msup><mo>:</mo><mi>sSet</mi><mo>→</mo><mi>Kan</mi></mrow><annotation encoding="application/x-tex">Ex^\infty : sSet \to Kan</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Kan</mi></mrow><annotation encoding="application/x-tex">Kan</annotation></semantics></math> is the category of <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a>.</li> </ul> <p>This may be found, for instance, in II.8.6–7 of <a href="model+structure+on+simplicial+sets#GoerssJardine">Goerss-Jardine</a>. Another proof may be found in <a href="model+structure+on+simplicial+sets#Moss">Moss</a>, and a different proof of properness may be found in <a href="model+structure+on+simplicial+sets#Cisinski06">Cisinski, Prop. 2.1.5</a>.</p> <h3 id="quillen_equivalence_with_">Quillen equivalence with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math></h3> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>The <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a>/<a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a>-<a class="existingWikiWord" href="/nlab/show/nerve+and+realization">adjunction</a> of example <a class="maruku-ref" href="#TopologicalRealizationOfSimplicialSets"></a> constitutes a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> of the classical model structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math> of def. <a class="maruku-ref" href="#ClassesOfMorphismsOnsSetQuillen"></a> with the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mo>⊣</mo><mi>Sing</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>Top</mi> <mi>Quillen</mi></msub><mo lspace="0em" rspace="thinmathspace">underoversey</mo><munder><mo>⟶</mo><mi>Sing</mi></munder><mover><mo>⟵</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mover><mrow><msub><mo>⊥</mo> <mpadded width="0"><mrow><msub><mrow></mrow> <mi>Qu</mi></msub></mrow></mpadded></msub></mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex"> ({\vert -\vert}\dashv Sing) \;\;\colon\;\; Top_{Quillen} \underoversey {\underset{Sing}{\longrightarrow}} {\overset{{\vert -\vert}}{\longleftarrow}} { \bot_{\mathrlap{{}_{Qu}}} } sSet_{Quillen} </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_9">Proof</h6> <p>First of all, the adjunction is indeed a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a>: prop. <a class="maruku-ref" href="#SingDetextsAndReflectsFibrations"></a> says in particular that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sing(-)</annotation></semantics></math> takes <a class="existingWikiWord" href="/nlab/show/Serre+fibrations">Serre fibrations</a> to <a class="existingWikiWord" href="/nlab/show/Kan+fibrations">Kan fibrations</a> and prop. <a class="maruku-ref" href="#TopologicalRealizationOfsSetLandsInCWComplexes"></a> gives that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{\vert-\vert}</annotation></semantics></math> sends monomorphisms of simplicial sets to <a class="existingWikiWord" href="/nlab/show/relative+cell+complexes">relative cell complexes</a>.</p> <p>Now prop. <a class="maruku-ref" href="#UnitOfSingularNerveAndRealizationIsWEOnKanComplexes"></a> says that the derived adjunction unit and counit are weak equivalences, and hence the Quillen adjunction is a Quillen equivalence.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/constructive+model+structure+on+simplicial+sets">constructive model structure on simplicial sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+reduced+simplicial+sets">model structure on reduced simplicial sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+adjunction+between+simplicial+sets+and+connective+dgc-algebras">Quillen adjunction between simplicial sets and connective dgc-algebras</a></p> </li> </ul> <h2 id="references">References</h2> <div> <h3 id="classical_model_structure_on_simplicial_sets">Classical model structure on simplicial sets</h3> <p>On the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a>:</p> <p>The original proof is due to</p> <ul> <li id="Quillen67"><a class="existingWikiWord" href="/nlab/show/Daniel+Quillen">Daniel Quillen</a>, Section II.3 of: <em>Axiomatic homotopy theory</em> in: <em><a class="existingWikiWord" href="/nlab/show/Homotopical+Algebra">Homotopical Algebra</a></em>, Lecture Notes in Mathematics 43, Springer 1967(<a href="https://doi.org/10.1007/BFb0097438">doi:10.1007/BFb0097438</a>)</li> </ul> <p>This proof is purely <a class="existingWikiWord" href="/nlab/show/combinatorics">combinatorial</a> (i.e. does not pass through <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> as <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>): Quillen uses the theory of <a class="existingWikiWord" href="/nlab/show/minimal+Kan+fibrations">minimal Kan fibrations</a>, the fact that the latter are <a class="existingWikiWord" href="/nlab/show/fiber+bundles">fiber bundles</a>, as well as the fact that the <a class="existingWikiWord" href="/nlab/show/simplicial+classifying+space">simplicial classifying space</a> of a <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a> is a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>.</p> <p>Other proofs are were given in:</p> <ul> <li id="GelfandManin96"> <p><a class="existingWikiWord" href="/nlab/show/Sergei+Gelfand">Sergei Gelfand</a>, <a class="existingWikiWord" href="/nlab/show/Yuri+Manin">Yuri Manin</a>, Sections V.1-2 of: <em><a class="existingWikiWord" href="/nlab/show/Methods+of+homological+algebra">Methods of homological algebra</a></em>, transl. from the 1988 Russian (Nauka Publ.) original, Springer 1996. xviii+372 pp. 2nd corrected ed. 2002 (<a href="https://doi.org/10.1007/978-3-662-12492-5">doi:10.1007/978-3-662-12492-5</a>)</p> </li> <li id="GoerssJardine99"> <p><a class="existingWikiWord" href="/nlab/show/Paul+Goerss">Paul Goerss</a>, <a class="existingWikiWord" href="/nlab/show/J.+F.+Jardine">J. F. Jardine</a>, Section I.11 of: <em><a class="existingWikiWord" href="/nlab/show/Simplicial+homotopy+theory">Simplicial homotopy theory</a></em>, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) (<a href="https://link.springer.com/book/10.1007/978-3-0346-0189-4">doi:10.1007/978-3-0346-0189-4</a>, <a href="http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/">webpage</a>)</p> </li> <li id="JoyalTierney08"> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Myles+Tierney">Myles Tierney</a>, <em>Notes on simplicial homotopy theory</em>, Lecture at <em><a href="https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html">Advanced Course on Simplicial Methods in Higher Categories</a></em>, CRM 2008 (<a class="existingWikiWord" href="/nlab/files/JoyalTierneyNotesOnSimplicialHomotopyTheory.pdf" title="pdf">pdf</a>)</p> </li> <li id="JoyalTierney09"> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Myles+Tierney">Myles Tierney</a> <em>An introduction to simplicial homotopy theory</em>, 2005 (<a href="http://hopf.math.purdue.edu/cgi-bin/generate?/Joyal-Tierney/JT-chap-01">web</a>, <a class="existingWikiWord" href="/nlab/files/JoyalTierneySimplicialHomotopyTheory.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>A proof (in fact two variants of it) using the <a class="existingWikiWord" href="/nlab/show/Kan+fibrant+replacement">Kan fibrant replacement</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ex</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">Ex^\infty</annotation></semantics></math> functor is given (in the context of_<a class="existingWikiWord" href="/nlab/show/Cisinski+model+structure">Cisinski model structure</a><em>) in:</em></p> <ul> <li id="Cisinski06"><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a>, Section 2 of: <em><a class="existingWikiWord" href="/joyalscatlab/published/Les+pr%C3%A9faisceaux+comme+type+d%27homotopie">Les préfaisceaux comme type d'homotopie</a></em>, Astérisque, Volume 308, Soc. Math. France (2006), 392 pages (<a href="http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf">pdf</a>)</li> </ul> <p>The crucial step is the proof that the fibrations are precisely the <a class="existingWikiWord" href="/nlab/show/Kan+fibrations">Kan fibrations</a> (and also to prove all the good properties of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ex</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">Ex^\infty</annotation></semantics></math> without using topological spaces); for two different proofs of this fact using <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ex</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">Ex^\infty</annotation></semantics></math>, see Prop. 2.1.41 as well as Scholium 2.3.21 for an alternative). For the rest, everything was already in the book of Gabriel and Zisman, for instance.</p> <p>Another approach using <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ex</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">Ex^\infty</annotation></semantics></math> is:</p> <ul> <li id="Moss">Sean Moss, <em>Another approach to the Kan-Quillen model structure</em>, Journal of Homotopy and Related Structures volume 15, pages 143–165 (2020) (<a href="http://arxiv.org/abs/1506.04887">arXiv:1506.04887</a>, <a href="https://doi.org/10.1007/s40062-019-00247-y">doi:10.1007/s40062-019-00247-y</a>)</li> </ul> <p>A proof of the model structure not relying on the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> nor on explicit models for <a class="existingWikiWord" href="/nlab/show/Kan+fibrant+replacement">Kan fibrant replacement</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Christian+Sattler">Christian Sattler</a>, <em>The Equivalence Extension Property and Model Structures</em> (<a href="https://arxiv.org/abs/1704.06911">arXiv:1704.06911</a>)</li> </ul> <p>Proofs valid in <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a> are given in:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Simon+Henry">Simon Henry</a>, <em>A constructive account of the Kan-Quillen model structure and of Kan’s Ex∞ functor</em>, <a href="https://arxiv.org/abs/1905.06160">arXiv:1905.06160</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nicola+Gambino">Nicola Gambino</a>, <a class="existingWikiWord" href="/nlab/show/Simon+Henry">Simon Henry</a>, <a class="existingWikiWord" href="/nlab/show/Christian+Sattler">Christian Sattler</a>, <a class="existingWikiWord" href="/nlab/show/Karol+Szumi%C5%82o">Karol Szumiło</a>, <em>The effective model structure and ∞-groupoid objects</em>, <a href="https://arxiv.org/abs/2102.06146">arXiv:2102.06146</a>.</p> </li> </ul> <p>As a <a href="relation+between+type+theory+and+category+theory#HomotopyTypeTheory">categorical semantics for homotopy type theory</a>, the model structure on simplicial sets is considered in</p> <ul> <li id="KapulkinLumnsdaineVoevodsky12"> <p><a class="existingWikiWord" href="/nlab/show/Chris+Kapulkin">Chris Kapulkin</a>, <a class="existingWikiWord" href="/nlab/show/Peter+LeFanu+Lumsdaine">Peter LeFanu Lumsdaine</a>, <a class="existingWikiWord" href="/nlab/show/Vladimir+Voevodsky">Vladimir Voevodsky</a>, (<a href="http://arxiv.org/abs/1203.2553">arXiv:1203.2553</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chris+Kapulkin">Chris Kapulkin</a>, <a class="existingWikiWord" href="/nlab/show/Peter+LeFanu+Lumsdaine">Peter LeFanu Lumsdaine</a>, <em>The Simplicial Model of Univalent Foundations (after Voevodsky)</em>, Journal of the European Mathematical Society (<a href="https://arxiv.org/abs/1211.2851">arXiv:1211.2851</a>,<a href="http://doi.org/10.4171/jems/1050">doi:10.4171/jems/1050</a>)</p> </li> </ul> </div></body></html> </div> <div class="revisedby"> <p> Last revised on June 20, 2022 at 10:25:52. 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