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href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</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/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="topology">Topology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topology">topology</a></strong> (<a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>, <a class="existingWikiWord" href="/nlab/show/point-free+topology">point-free topology</a>)</p> <p>see also <em><a class="existingWikiWord" 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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#OfSimplicialSets'>Of cell complexes such as simplicial sets</a></li> <li><a href='#OfSimplicialTopologicalSpaces'>Of simplicial topological spaces</a></li> <li><a href='#OfCohesiveInfinityGroupoids'>Of cohesive <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</a></li> <li><a href='#OfSimplicialObjects'>Of simplicial objects in a category</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#realizations_as_cw_complexes'>Realizations as CW complexes</a></li> <li><a href='#GeometricRealizationIsLeftExact'>Theorem: Geometric realization is left exact</a></li> <ul> <li><a href='#geometric_realization_preserves_equalizers'>Geometric realization preserves equalizers</a></li> <li><a href='#geometric_realization_preserves_finite_products'>Geometric realization preserves finite products</a></li> </ul> <li><a href='#a_construction_of_drinfeld_of_geometric_realization_as_hom01'>A construction of Drinfeld of geometric realization as Hom([0,1],-)</a></li> <li><a href='#geometric_realization_preserves_fibrations'>Geometric realization preserves fibrations</a></li> <li><a href='#induced_properties_of_the_fibrant_replacement'>Induced properties of the fibrant replacement</a></li> <li><a href='#geometric_realization_of_barycentric_subdivisions'>Geometric realization of barycentric subdivisions</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#compatibility_with_homotopy_limits'>Compatibility with homotopy limits</a></li> <li><a href='#geometric_realization_of_barycentric_subdivisions_2'>Geometric realization of barycentric subdivisions</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>What is commonly called <em>geometric realization</em> (or, less commonly but more accurately: <em>topological realization</em>,see Rem. <a class="maruku-ref" href="#OnTerminology"></a>) is the operation that builds from 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>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> 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><mo stretchy="false">|</mo><mi>X</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|X|</annotation></semantics></math> obtained by interpreting each element 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> – each abstract <a class="existingWikiWord" href="/nlab/show/n-simplex"><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</a> 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> – as one copy of 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 <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></mrow><annotation encoding="application/x-tex">\Delta^n_{Top}</annotation></semantics></math> and then gluing together all these along their <a class="existingWikiWord" href="/nlab/show/boundaries">boundaries</a> to a big topological space, using the information encoded in the <a class="existingWikiWord" href="/nlab/show/face+map">face and degeneracy maps</a> 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> on how these simplices are supposed to be stuck together. This procedure generalises the geometric realization of <a class="existingWikiWord" href="/nlab/show/simplicial+complexes">simplicial complexes</a> as described at that entry.</p> <p>Geometric realization is the special case of the general notion of <a class="existingWikiWord" href="/nlab/show/nerve+and+realization">nerve and realization</a> that is induced from the standard <a class="existingWikiWord" href="/nlab/show/simplicial+set">cosimplicial</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><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↦</mo><msubsup><mi>Δ</mi> <mi>Top</mi> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">[n] \mapsto \Delta^n_{Top}</annotation></semantics></math>. (N.B.: in this article, <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> denotes the ordinal with <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> elements. The corresponding contravariant representable is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta(-, n)</annotation></semantics></math>.) Analogous constrtuctions yield <a class="existingWikiWord" href="/nlab/show/cubical+geometric+realization">cubical geometric realization</a>, etc.</p> <p>In the context of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> geometric realization plays a notable role in the <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>, where it is part of the <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> between the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> and the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a>.</p> <p>The construction generalizes naturally to a map from <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial</a> <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> to plain topological spaces. For more on that see <em><a class="existingWikiWord" href="/nlab/show/geometric+realization+of+simplicial+spaces">geometric realization of simplicial spaces</a></em>.</p> <p>The dual concept is <em><a class="existingWikiWord" href="/nlab/show/totalization">totalization</a></em> .</p> <p> <div class='num_remark' id='OnTerminology'> <h6>Remark</h6> <p><strong>(alternative terminology: “topological realization”)</strong> <br /> While the term <em>geometric realization</em> is classical (<a href="#Quillen68">Quillen 1968</a>) and in common use by <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topologists</a> (e.g. <a href="#GoerssJardine09">Goerss & Jardine 2009, §VII.3</a>), it is somewhat inaccurate or at least misleading, in that the bare <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> produced by the realization construction are not yet what <a class="existingWikiWord" href="/nlab/show/geometry">geometers</a> would commonly regard as reflecting <a class="existingWikiWord" href="/nlab/show/geometry">geometric</a> <a class="existingWikiWord" href="/nlab/show/mathematical+structure">structure</a>: <a class="existingWikiWord" href="/nlab/show/geometry">Geometry</a> on a topological space is instead typically taken to be the <a class="existingWikiWord" href="/nlab/show/extra+structure">extra structure</a> of a <a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a> (e.g. <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a>) together with some kind of <a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a> (such as <a class="existingWikiWord" href="/nlab/show/Riemannian+structure">Riemannian structure</a>, <a class="existingWikiWord" href="/nlab/show/conformal+structure">conformal structure</a>, etc.). This terminological mismatch is quite stark in the main application to <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, where one cares only about the (<a class="existingWikiWord" href="/nlab/show/weak+homotopy+type">weak</a>) <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of the realization, which is the most non-geometric aspect of any notion of “<a class="existingWikiWord" href="/nlab/show/space">space</a>”. This becomes a real issue in contexts of <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a> and more generally of <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric homotopy theory</a> (such as for <a class="existingWikiWord" href="/nlab/show/differential+cohomology+theories">differential cohomology theories</a>) where realizations of simplicial sets to genuine geometric spaces richer than topological spaces do play a role (e.g. already for the notion of smooth/<a class="existingWikiWord" href="/nlab/show/equivariant+triangulation">equivariant</a> <a class="existingWikiWord" href="/nlab/show/triangulations">triangulations</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>).</p> <p>In conclusion, a more accurate and more descriptive term for the realization operation discussed here would be <strong>topological realization</strong>. References which use this terminology include <a href="#Simpson96">Simpson 1996</a>, <a href="simplicial+approximation+theorem#Jardine04">Jardine 2004</a>, <a href="#Lackenby08">Lackenby 2008, §I.2</a>. In <a href="operad#FresseHOGTG">Fresse 2017</a> the term “topological realization” is used in the pdf draft but is replaced by “geometric realization” in the published version.</p> </div> </p> <h2 id="definition">Definition</h2> <p>There are various levels of generality in which the notion of (topological) geometric realization makes sense. The basic definition is</p> <ul> <li><a href="#OfSimplicialSets">For cell complexes such as simplicial sets</a>.</li> </ul> <p>A generalization of this of central importance is the</p> <ul> <li><a href="#OfSimplicialTopologicalSpaces">Geometric realization of simplicial topological spaces</a></li> </ul> <p>Up to homotopy, this is a special case of a general notion of</p> <ul> <li><a href="#OfCohesiveInfinityGroupoids">Geometric realization of cohesive ∞-groupoids</a>.</li> </ul> <p>At the point-set level, it is also a special case of a general notion of</p> <ul> <li><a href="#OfSimplicialObjects">Geometric realization of simplicial objects</a>.</li> </ul> <h3 id="OfSimplicialSets">Of cell complexes such as simplicial sets</h3> <p>Let <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> be one of the categories of <a class="existingWikiWord" href="/nlab/show/geometric+shapes+for+higher+structures">geometric shapes for higher structures</a>, such as the <a class="existingWikiWord" href="/nlab/show/globe+category">globe category</a> or the <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a> or the <a class="existingWikiWord" href="/nlab/show/cube+category">cube category</a>.</p> <p>There is an obvious functor</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>st</mi><mo>:</mo><mi>S</mi><mo>→</mo></mrow><annotation encoding="application/x-tex"> st : S \to </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> <p>which sends the standard <em>cellular</em> shape <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> (the standard cellular <a class="existingWikiWord" href="/nlab/show/globe">globe</a>, <a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> or <a class="existingWikiWord" href="/nlab/show/cube">cube</a>, respectively) to the corresponding standard <em>topological</em> shape (for instance the standard <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>st</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msubsup><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></msubsup><msub><mi>x</mi> <mi>i</mi></msub><mo>=</mo><mn>1</mn><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> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> st([n]) := \{ (x_1, \cdots, x_n) | \sum_{i=1}^n x_i = 1, x_i \geq 0 \} \subset \mathbb{R}^{n}</annotation></semantics></math> ) with the obvious induced face and boundary maps.</p> <p>Using this, in cases 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> can be regarded as <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> over and <a class="existingWikiWord" href="/nlab/show/copower">tensored</a> over a base category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, the <strong>geometric realization</strong> of a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mo>•</mo></msup><mo>:</mo><msup><mi>S</mi> <mi>op</mi></msup><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">K^\bullet : S^{op} \to V</annotation></semantics></math> on <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> – e.g., of a <a class="existingWikiWord" href="/nlab/show/globular+set">globular set</a>, a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> or a <a class="existingWikiWord" href="/nlab/show/cubical+set">cubical set</a>, respectively (when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">V= Set</annotation></semantics></math>) – is the <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> given by the <a class="existingWikiWord" href="/nlab/show/end">coend</a>, <a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted colimit</a>, or <a class="existingWikiWord" href="/nlab/show/tensor+product+of+functors">tensor product of functors</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><msup><mi>K</mi> <mo>•</mo></msup><mo stretchy="false">|</mo><mo>=</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>S</mi></mrow></msup><mi>st</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>⋅</mo><msup><mi>K</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> |K^\bullet| = \int^{[n] \in S} st([n]) \cdot K^n \,. </annotation></semantics></math></div> <p>In the case of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>, see for more discussion also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/nerve">nerve</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>.</li> </ul> <p>Via simplicial <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> functors geometric realization of simplicial sets induces geometric realizations of many other structures, for instance</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+realization+of+categories">geometric realization of categories</a></li> </ul> <p>For the case of <a class="existingWikiWord" href="/nlab/show/cubical+set">cubical sets</a>, see <a class="existingWikiWord" href="/nlab/show/cubical+geometric+realisation">cubical geometric realisation</a>.</p> <h3 id="OfSimplicialTopologicalSpaces">Of simplicial topological spaces</h3> <p>See</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+realization+of+simplicial+topological+spaces">geometric realization of simplicial topological spaces</a></li> </ul> <h3 id="OfCohesiveInfinityGroupoids">Of cohesive <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</h3> <p>Every <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> (in fact every <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a>) comes with its intrinsic notion of geometric realization.</p> <p>The general abstract definition is at <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> in the section <a href="http://nlab.mathforge.org/nlab/show/cohesive+%28infinity%2C1%29-topos%20--%20structures#Homotopy">Geometric homotopy</a>.</p> <p>For the choice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> this reproduces the <a href="#OfSimplicialSets">geometric realization of simplicial sets</a>, see at <a class="existingWikiWord" href="/nlab/show/discrete+%E2%88%9E-groupoid">discrete ∞-groupoid</a> the section <a href=""></a></p> <p>For the choice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/ETop%E2%88%9EGrpd">ETop∞Grpd</a> and <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> this reproduces <a href="#OfSimplicialTopologicalSpaces">geometric realization of simplicial topological spaces</a>. See the sections <a href="http://nlab.mathforge.org/nlab/show/Euclidean-topological+infinity-groupoid#GeometricHomotopy">ETop∞Grpd – Geometric homotopy</a> and <a href="http://nlab.mathforge.org/nlab/show/smooth+infinity-groupoid#StrucGeometricHomotopy">Smooth ∞-groupoid – Geometric homotopy</a></p> <h3 id="OfSimplicialObjects">Of simplicial objects in a category</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/cocomplete+category">cocomplete</a> <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a> with <a class="existingWikiWord" href="/nlab/show/copowers">copowers</a>. A <strong><a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a></strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">X:\Delta^{op}\to M</annotation></semantics></math>, where <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/simplex+category">simplex category</a>. Its <strong>geometric realization</strong> is defined similarly to the classical case as a <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><mo stretchy="false">|</mo><mi>X</mi><mo stretchy="false">|</mo><mo>=</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⊙</mo><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> |X| = \int^{[n]\in\Delta} \Delta[n] \odot X_n </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊙</mo></mrow><annotation encoding="application/x-tex">\odot</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/copower">copower</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>. This operation is a left adjoint which is even a simplicially enriched functor; see <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a> for more details.</p> <h2 id="properties">Properties</h2> <p>In this section we consider topological <a href="#OfSimplicialSets">geometric realization of simplicial sets</a>, which is the best studied and perhaps most significant case.</p> <h3 id="realizations_as_cw_complexes">Realizations as CW complexes</h3> <p>Each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>X</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{|X|}</annotation></semantics></math> is a CW complex (see lemma <a class="maruku-ref" href="#mono"></a> below), and so geometric realization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo>:</mo><msup><mi>Set</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">{|(-)|}: Set^{\Delta^{op}} \to Top</annotation></semantics></math> takes values in the full subcategory of CW complexes, and therefore in any <a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a>, for example in the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CGHaus</mi></mrow><annotation encoding="application/x-tex">CGHaus</annotation></semantics></math> of compactly generated Hausdorff spaces. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Space</mi></mrow><annotation encoding="application/x-tex">Space</annotation></semantics></math> be any convenient category of topological spaces, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>Space</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">i \colon Space \to Top</annotation></semantics></math> denote the inclusion.</p> <div class="un_prop"> <h6 id="proposition">Proposition</h6> <p>For any simplicial set <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>, there is a natural isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><msup><mo>∫</mo> <mrow><mi>n</mi><mo>:</mo><mi>Δ</mi></mrow></msup><mi>X</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≅</mo><mrow><mo stretchy="false">|</mo><mi>X</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">i(\int^{n: \Delta} X(n) \cdot \sigma(n)) \cong {|X|}</annotation></semantics></math>, where the coend on the left is computed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Space</mi></mrow><annotation encoding="application/x-tex">Space</annotation></semantics></math>.</p> </div> <p>This is obvious: more generally, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>J</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">F: J \to A</annotation></semantics></math> is a diagram and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>A</mi><mo>↪</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">i: A \hookrightarrow B</annotation></semantics></math> is a full replete subcategory, and if the colimit in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∘</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">i \circ F</annotation></semantics></math> lands in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, then this is also the colimit 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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. (The dual statement also holds, with limits instead of colimits.)</p> <p>Below, we let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>:</mo><msup><mi>Set</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mo>→</mo><mi>Space</mi></mrow><annotation encoding="application/x-tex">R: Set^{\Delta^{op}} \to Space</annotation></semantics></math> denote the geometric realization when considered as landing in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Space</mi></mrow><annotation encoding="application/x-tex">Space</annotation></semantics></math>.</p> <h3 id="GeometricRealizationIsLeftExact">Theorem: Geometric realization is left exact</h3> <p>We continue to assume <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Space</mi></mrow><annotation encoding="application/x-tex">Space</annotation></semantics></math> is any <a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a>. In this section we prove that geometric realization</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>:</mo><msup><mi>Set</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mo>→</mo><mi>Space</mi></mrow><annotation encoding="application/x-tex">R: Set^{\Delta^{op}} \to Space</annotation></semantics></math></div> <p>is a left <a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a> in that it preserves <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a>.</p> <p>It is important that we use some such “convenience” assumption, because for example</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 stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo>:</mo><msup><mi>Set</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mo>→</mo><mi>Top</mi><mo>,</mo></mrow><annotation encoding="application/x-tex">{|(-)|}: Set^{\Delta^{op}} \to Top,</annotation></semantics></math></div> <p>valued in general <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, does not preserve products. (To get a correct statement, one usual procedure is to “kelley-fy” products by applying the coreflection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo lspace="verythinmathspace">:</mo><mi>Haus</mi><mo>→</mo><mi>CGHaus</mi></mrow><annotation encoding="application/x-tex">k \colon Haus \to CGHaus</annotation></semantics></math> to <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff</a> and <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a>. This gives the correct isomorphism in the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Space</mi><mo>=</mo><mi>CGHaus</mi></mrow><annotation encoding="application/x-tex">Space = CGHaus</annotation></semantics></math>, where we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">|</mo></mrow><mo>≅</mo><mrow><mo stretchy="false">|</mo><mi>X</mi><mo stretchy="false">|</mo></mrow><msub><mo>×</mo> <mi>k</mi></msub><mrow><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo></mrow><mo>≔</mo><mi>k</mi><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mi>X</mi><mo stretchy="false">|</mo></mrow><mo>×</mo><mrow><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">{|X \times Y|} \cong {|X|} \times_k {|Y|} \coloneqq k({|X|} \times {|Y|})</annotation></semantics></math>; the product on the right has been “kelleyfied” to the product appropriate for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CGHaus</mi></mrow><annotation encoding="application/x-tex">CGHaus</annotation></semantics></math>.)</p> <p>We reiterate that <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> denotes the geometric realization functor considered as valued in a convenient category of spaces, whereas <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{|(-)|}</annotation></semantics></math> is geometric realization viewed as taking values 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> <div class="num_theorem" id="leftexact"> <h6 id="theorem">Theorem</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>=</mo><mi>hom</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>Space</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">U = \hom(1, -): Space \to Set</annotation></semantics></math> be the underlying-set functor. Then the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mi>R</mi><mo>:</mo><msup><mi>Set</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">U R: Set^{\Delta^{op}} \to Set</annotation></semantics></math> is left exact.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>As described at the nLab article on triangulation <a href="triangulation#StandardAffineSimplexFunctor">here</a>, the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mover><mo>→</mo><mi>σ</mi></mover><mi>Space</mi><mover><mo>→</mo><mi>U</mi></mover><mi>Set</mi></mrow><annotation encoding="application/x-tex">\Delta \stackrel{\sigma}{\to} Space \stackrel{U}{\to} Set</annotation></semantics></math></div> <p>can be described as the functor</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>≅</mo><msup><mi>FinInt</mi> <mi>op</mi></msup><mo>↪</mo><msup><mi>Int</mi> <mi>op</mi></msup><mover><mo>→</mo><mrow><mi>Int</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo></mrow></mover><mi>Set</mi></mrow><annotation encoding="application/x-tex">\Delta \cong FinInt^{op} \hookrightarrow Int^{op} \stackrel{Int(-, I)}{\to} Set</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Int</mi></mrow><annotation encoding="application/x-tex">Int</annotation></semantics></math> is the category of intervals (linearly ordered sets with distinct top and bottom). Because every interval, in particular <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 <a class="existingWikiWord" href="/nlab/show/filtered+colimit">filtered colimit</a> of finite intervals, and because finite intervals are finitely presentable intervals, it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mi>σ</mi><mo lspace="verythinmathspace">:</mo><mi>Δ</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">U \sigma \colon \Delta \to Set</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/flat+functor">flat functor</a> (a <a class="existingWikiWord" href="/nlab/show/filtered+colimit">filtered colimit</a> of representables). But on general grounds, <a class="existingWikiWord" href="/nlab/show/copower">tensoring</a> with a flat functor is left exact, which in this case means</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mi>R</mi><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo>⊗</mo> <mi>Δ</mi></msub><mi>U</mi><mi>σ</mi><mo>:</mo><msup><mi>Set</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">U R = - \otimes_\Delta U \sigma: Set^{\Delta^{op}} \to Set</annotation></semantics></math></div> <p>is left exact.</p> </div> <p>Obviously the preceding proof is not sensitive to whether we use <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Space</mi></mrow><annotation encoding="application/x-tex">Space</annotation></semantics></math> or <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> <h4 id="geometric_realization_preserves_equalizers">Geometric realization preserves equalizers</h4> <div class="num_lemma" id="mono"> <h6 id="lemma">Lemma</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">i: X \to Y</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> of simplicial sets, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>:</mo><mi>R</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>R</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(i): R(X) \to R(Y)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed subspace</a> inclusion, in fact a <a class="existingWikiWord" href="/nlab/show/relative+CW-complex">relative CW-complex</a>. In particular, taking <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">X = \emptyset</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(Y)</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CW</mi></mrow><annotation encoding="application/x-tex">CW</annotation></semantics></math>-complex.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Any monomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">i \colon X \to Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">Set^{\Delta^{op}}</annotation></semantics></math> can be seen as the result of iteratively adjoining nondegenerate <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 other words, there is a chain of inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>↪</mo><mi>F</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>↪</mo><mi>…</mi><mi>Y</mi><mo>=</mo><msub><mi>colim</mi> <mi>i</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X = F(0) \hookrightarrow F(1) \hookrightarrow \ldots Y = colim_i F(i)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>κ</mi><mo>→</mo><msup><mi>Set</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">F: \kappa \to Set^{\Delta^{op}}</annotation></semantics></math> is a functor from some ordinal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo>≤</mo><mn>1</mn><mo>≤</mo><mi>…</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\kappa = \{0 \leq 1\leq \ldots\}</annotation></semantics></math> (as <a class="existingWikiWord" href="/nlab/show/preorder">preorder</a>) that preserves directed colimits, and each inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>α</mi><mo>≤</mo><mi>α</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>:</mo><mi>F</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>α</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(\alpha \leq \alpha + 1): F(\alpha) \to F(\alpha + 1)</annotation></semantics></math> fits into a pushout diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>∂</mo><mi>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mpadded width="0" lspace="-100%width"><mi>j</mi></mpadded><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>α</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \partial \Delta(-, n) & \to & F(\alpha) \\ \mathllap{j} \downarrow & & \downarrow \\ \Delta(-, n) & \to & F(\alpha+1) } </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> is the inclusion. Now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(j)</annotation></semantics></math> is identifiable as the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^{n-1} \to D^n</annotation></semantics></math>, and since <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> preserves pushouts (which are calculated as they are 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>), we see by <a href="/nlab/show/subspace+topology#pushout">this lemma</a> that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>F</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>→</mo><mi>R</mi><mi>F</mi><mo stretchy="false">(</mo><mi>α</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R F(\alpha) \to R F(\alpha+1)</annotation></semantics></math> is a closed subspace inclusion and evidently a <a class="existingWikiWord" href="/nlab/show/relative+CW-complex">relative CW-complex</a>. By <a href="/nlab/show/subspace+topology#transfinite">another lemma</a>, it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \to Y</annotation></semantics></math> is also a closed inclusion and indeed a relative CW-complex.</p> </div> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>:</mo><msup><mi>Set</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mo>→</mo><mi>Space</mi></mrow><annotation encoding="application/x-tex">R: Set^{\Delta^{op}} \to Space</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/equalizers">equalizers</a>.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>The equalizer of a pair of maps 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> is computed as the equalizer on the level of underlying sets, equipped with the subspace topology. So if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mover><mo>→</mo><mi>i</mi></mover><mi>X</mi><mover><munder><mo>⟶</mo><mi>g</mi></munder><mover><mo>⟶</mo><mi>f</mi></mover></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">E \stackrel{i}{\to} X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} Y</annotation></semantics></math></div> <p>is an equalizer diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">Set^{\Delta^{op}}</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>i</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{|i|}</annotation></semantics></math> is the equalizer of the pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">|</mo></mrow></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><mrow><mo stretchy="false">|</mo><mi>g</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{|g|}</annotation></semantics></math>, because the underlying function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mi>i</mi><mo stretchy="false">|</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U({|i|})</annotation></semantics></math> is the equalizer of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">|</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U({|f|})</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mi>g</mi><mo stretchy="false">|</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U({|g|})</annotation></semantics></math> on the underlying set level by the preceding theorem, and because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>i</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{|i|}</annotation></semantics></math> is a (closed) subspace inclusion by lemma <a class="maruku-ref" href="#mono"></a>. But this <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>-equalizer <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>i</mi><mo stretchy="false">|</mo></mrow><mo>:</mo><mrow><mo stretchy="false">|</mo><mi>E</mi><mo stretchy="false">|</mo></mrow><mo>→</mo><mrow><mo stretchy="false">|</mo><mi>X</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{{|i|}}: {{|E|}} \to {{|X|}}</annotation></semantics></math> lives in the full subcategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Space</mi></mrow><annotation encoding="application/x-tex">Space</annotation></semantics></math>, and therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo stretchy="false">|</mo><mi>i</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">R(i) = {|i|}</annotation></semantics></math> is the equalizer of the pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">R(f) = {|f|}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo stretchy="false">|</mo><mi>g</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">R(g) = {|g|}</annotation></semantics></math>.</p> </div> <p>As the proof indicates, that realization preserves equalizers is not at all sensitive to whether we use <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> or a convenient category of spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Space</mi></mrow><annotation encoding="application/x-tex">Space</annotation></semantics></math>.</p> <h4 id="geometric_realization_preserves_finite_products">Geometric realization preserves finite products</h4> <p>That geometric realization preserves products <em>is</em> sensitive to whether we think of it as valued 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> or in a convenient category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Space</mi></mrow><annotation encoding="application/x-tex">Space</annotation></semantics></math>. In particular, the proof uses <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><mi>Space</mi></mrow><annotation encoding="application/x-tex">Space</annotation></semantics></math> in an essential way (in the form that <a class="existingWikiWord" href="/nlab/show/finite+products">finite products</a> distribute over arbitrary <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>).</p> <p>First, an easy result on products of simplices.</p> <div class="num_lemma" id="product"> <h6 id="lemma_2">Lemma</h6> <p>The realization of a product of two representables <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>m</mi><mo stretchy="false">)</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta(-, m) \times \Delta(-, n)</annotation></semantics></math> is compact.</p> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>It suffices to observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>m</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">\Delta[m] \times \Delta[n]</annotation></semantics></math> has finitely many non-degenerate simplices. That is clear since non-degenerate <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>-simplices in the nerve of a poset <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> are exactly injective order preserving maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">[k] \to P</annotation></semantics></math>.</p> </div> <div class="num_lemma" id="canonical"> <h6 id="lemma_3">Lemma</h6> <p>The canonical map</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>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>m</mi><mo stretchy="false">)</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>m</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo>×</mo><mrow><mo stretchy="false">|</mo><mi>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{|\Delta(-, m) \times \Delta(-, n)|} \to {|\Delta(-, m)|} \times {|\Delta(-, n)|}</annotation></semantics></math></div> <p>is a homeomorphism.</p> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>The canonical map is continuous, and a bijection at the underlying set level by theorem <a class="maruku-ref" href="#leftexact"></a>. The codomain is the compact Hausdorff space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>×</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sigma(m) \times \sigma(n)</annotation></semantics></math>, and the domain is compact by Lemma <a class="maruku-ref" href="#product"></a>. But a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The key properties of <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> needed for this subsection are (1) the fact it is compact Hausdorff, and (2) the order relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\leq</annotation></semantics></math> on the interval <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> defines a closed subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">I \times I</annotation></semantics></math>. These properties ensure that the affine <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>I</mi> <mi>n</mi></msup><mo>:</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>≤</mo><mi>…</mi><mo>≤</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{(x_1, \ldots, x_n) \in I^n: x_1 \leq \ldots \leq x_n\}</annotation></semantics></math> is itself compact Hausdorff, so that the proof of lemma <a class="maruku-ref" href="#canonical"></a> goes through. The point is that in place of <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>, we can really use any interval <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> that satisfies these properties, thus defining an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>-based geometric realization instead of the standard (<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>-based) geometric realization being developed here.</p> </div> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>:</mo><msup><mi>Set</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mo>→</mo><mi>Space</mi></mrow><annotation encoding="application/x-tex">R: Set^{\Delta^{op}} \to Space</annotation></semantics></math> preserves products.</p> </div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>The proof is purely formal. Let <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> and <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> be simplicial sets. By the <a class="existingWikiWord" href="/nlab/show/co-Yoneda+lemma">co-Yoneda lemma</a>, we have isomorphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>≅</mo><msup><mo>∫</mo> <mi>m</mi></msup><mi>X</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>m</mi><mo stretchy="false">)</mo><mspace width="2em"></mspace><mi>Y</mi><mo>≅</mo><msup><mo>∫</mo> <mi>n</mi></msup><mi>Y</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \cong \int^m X(m) \cdot \Delta(-, m) \qquad Y \cong \int^n Y(n) \cdot \Delta(-, n)</annotation></semantics></math></div> <p>and so we calculate</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>R</mi><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≅</mo></mtd> <mtd><mi>R</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msup><mo>∫</mo> <mi>m</mi></msup><mi>X</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>m</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><msup><mo>∫</mo> <mi>n</mi></msup><mi>Y</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≅</mo></mtd> <mtd><mi>R</mi><mo stretchy="false">(</mo><msup><mo>∫</mo> <mi>m</mi></msup><msup><mo>∫</mo> <mi>n</mi></msup><mi>X</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>m</mi><mo stretchy="false">)</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≅</mo></mtd> <mtd><msup><mo>∫</mo> <mi>m</mi></msup><msup><mo>∫</mo> <mi>n</mi></msup><mi>X</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>R</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>m</mi><mo stretchy="false">)</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≅</mo></mtd> <mtd><msup><mo>∫</mo> <mi>m</mi></msup><msup><mo>∫</mo> <mi>n</mi></msup><mi>X</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>m</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo><mi>R</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≅</mo></mtd> <mtd><msup><mo>∫</mo> <mi>m</mi></msup><msup><mo>∫</mo> <mi>n</mi></msup><mi>X</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>×</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≅</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mo>∫</mo> <mi>m</mi></msup><mi>X</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><msup><mo>∫</mo> <mi>n</mi></msup><mi>Y</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≅</mo></mtd> <mtd><mi>R</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>×</mo><mi>R</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ R(X \times Y) & \cong & R((\int^m X(m) \cdot \Delta(-, m)) \times (\int^n Y(n) \cdot \Delta(-, n))) \\ & \cong & R(\int^m \int^n X(m) \cdot Y(n) \cdot (\Delta(-, m) \times \Delta(-, n))) \\ & \cong & \int^m \int^n X(m) \cdot Y(n) \cdot R(\Delta(-, m) \times \Delta(-, n)) \\ & \cong & \int^m \int^n X(m) \cdot Y(n) \cdot (R(\Delta(-, m)) \times R(\Delta(-, n)) \\ & \cong & \int^m \int^n X(m) \cdot Y(n) \cdot (\sigma(m) \times \sigma(n)) \\ & \cong & (\int^m X(m) \cdot \sigma(m)) \times (\int^n Y(n) \cdot \sigma(n)) \\ & \cong & R(X) \times R(Y) } </annotation></semantics></math></div> <p>where in each of the second and penultimate lines, we twice used the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>×</mo><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow><annotation encoding="application/x-tex">- \times -</annotation></semantics></math> preserves colimits in its separate arguments (i.e., the fact that the nice category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Space</mi></mrow><annotation encoding="application/x-tex">Space</annotation></semantics></math> is cartesian closed), and the remaining lines used the fact that <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> preserves colimits, and also products of representables by lemma <a class="maruku-ref" href="#canonical"></a>.</p> </div> <ul> <li>A slightly higher-level rendition of the proof might look like this:<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>R</mi><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≅</mo></mtd> <mtd><mi>R</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><msub><mo>⊗</mo> <mi>Δ</mi></msub><mi>hom</mi><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>Y</mi><msub><mo>⊗</mo> <mi>Δ</mi></msub><mi>hom</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≅</mo></mtd> <mtd><mi>R</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mrow><mi>Δ</mi><mo>×</mo><mi>Δ</mi></mrow></msub><mo stretchy="false">(</mo><mi>hom</mi><mo>×</mo><mi>hom</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≅</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mrow><mi>Δ</mi><mo>×</mo><mi>Δ</mi></mrow></msub><mi>R</mi><mo stretchy="false">(</mo><mi>hom</mi><mo>×</mo><mi>hom</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≅</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mrow><mi>Δ</mi><mo>×</mo><mi>Δ</mi></mrow></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>hom</mi><mo stretchy="false">)</mo><mo>×</mo><mi>R</mi><mo stretchy="false">(</mo><mi>hom</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≅</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>X</mi><msub><mo>⊗</mo> <mi>Δ</mi></msub><mi>R</mi><mo stretchy="false">(</mo><mi>hom</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>Y</mi><msub><mo>⊗</mo> <mi>Δ</mi></msub><mi>R</mi><mo stretchy="false">(</mo><mi>hom</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≅</mo></mtd> <mtd><mi>R</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo>⊗</mo> <mi>Δ</mi></msub><mi>hom</mi><mo stretchy="false">)</mo><mo>×</mo><mi>R</mi><mo stretchy="false">(</mo><mi>Y</mi><msub><mo>⊗</mo> <mi>Δ</mi></msub><mi>hom</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≅</mo></mtd> <mtd><mi>R</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>×</mo><mi>R</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ R(X \times Y) & \cong & R((X \otimes_{\Delta} \hom) \times (Y \otimes_{\Delta} \hom)) \\ & \cong & R((X \times Y) \otimes_{\Delta \times \Delta} (\hom \times \hom)) \\ & \cong & (X \times Y) \otimes_{\Delta \times \Delta} R(\hom \times \hom) \\ & \cong & (X \times Y) \otimes_{\Delta \times \Delta} (R(\hom) \times R(\hom)) \\ & \cong & (X \otimes_{\Delta} R(\hom)) \times (Y \otimes_{\Delta} R(\hom)) \\ & \cong & R(X \otimes_{\Delta} \hom) \times R(Y \otimes_{\Delta} \hom) \\ & \cong & R(X) \times R(Y) } </annotation></semantics></math></div></li> </ul> <h3 id="a_construction_of_drinfeld_of_geometric_realization_as_hom01">A construction of Drinfeld of geometric realization as Hom([0,1],-)</h3> <p><a href="https://arxiv.org/abs/math/0304064">Drinfeld, On the notion of geometric realization</a> provides a conceptual explanation of preserving finite limits, and “reformulates the definitions so that the following facts become obvious:</p> <ul> <li> <p>geometric realization commutes with finite projective limits (e.g., with Cartesian products);</p> </li> <li> <p>the geometric realization of a simplicial set … (resp. cyclic set) is equipped with an action of the group of orientation preserving homeomorphisms of the segment <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>:</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">I := [0, 1]</annotation></semantics></math> … (resp. the circle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^ 1</annotation></semantics></math>).“ (quote from the paper)</p> </li> </ul> <p>A draft <a href="http://mishap.sdf.org/Skorokhod_Geometric_Realisation.pdf">M.Gavrilovich, K.Pimenov. Geometric realisation as a Skorokhod semi-continuous path space endofunctor</a> attempts to further reformulate this by showing that, in a certain precise sense, geometric realisation is an endofunctor of a certain category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sF</mi></mrow><annotation encoding="application/x-tex">sF</annotation></semantics></math> of simplicial sets equipped with extra structure of topological nature (a notion of smallness). The underlying endofunctor of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSets</mi></mrow><annotation encoding="application/x-tex">sSets</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>HHom</mi><mo stretchy="false">(</mo><msub><mi>Hom</mi> <mi>preorders</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msub><mo stretchy="false">]</mo> <mo>≤</mo></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>Y</mi> <mo>.</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">HHom ( Hom_{preorders}(-, [0,1]_\leq), Y_.) </annotation></semantics></math></div> <p>The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sF</mi></mrow><annotation encoding="application/x-tex">sF</annotation></semantics></math> contains simplicial sets, topological and uniform spaces as full subcategories, and has forgetful functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sF</mi><mo>→</mo><mi>sSets</mi></mrow><annotation encoding="application/x-tex">sF\to sSets</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sF</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">sF\to Top</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sF</mi><mo>→</mo><mi>UniformSpaces</mi></mrow><annotation encoding="application/x-tex">sF\to UniformSpaces</annotation></semantics></math> such that the following compositions are identity: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSets</mi><mo>→</mo><mi>sF</mi><mo>→</mo><mi>sSets</mi></mrow><annotation encoding="application/x-tex">sSets\to sF\to sSets</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi><mo>→</mo><mi>sF</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top\to sF\to Top</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>UniformSpaces</mi><mo>→</mo><mi>sF</mi><mo>→</mo><mi>UniformSpaces</mi></mrow><annotation encoding="application/x-tex">UniformSpaces\to sF\to UniformSpaces</annotation></semantics></math>. Moreover, this endofunctor seems to have the right adjoint, defined by the usual construction.</p> <p>Here are some details. The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sF</mi></mrow><annotation encoding="application/x-tex">sF</annotation></semantics></math> may be thought as the category of simplicial sets with extra structure of topological nature, a notion of smallness. Formally it is just the category of simplicial objects in the category of <a class="existingWikiWord" href="/nlab/show/filters">filters</a>. The endofunctor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>HHom</mi><mo stretchy="false">(</mo><msub><mi>Hom</mi> <mi>preorders</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msub><mo stretchy="false">]</mo> <mo>≤</mo></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>Y</mi> <mo>.</mo></msub><mo stretchy="false">)</mo><mo>:</mo><mi>sF</mi><mo>→</mo><mi>sF</mi></mrow><annotation encoding="application/x-tex">HHom ( Hom_{preorders}(-, [0,1]_\leq), Y_.):sF\to sF </annotation></semantics></math> above is the inner hom of sSets equipped with an extra structure motivated by Skorokhod/Levi-Prokhorov convergence. The precise claim is that the geometric realisation of sSets factors as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>sSets</mi><mo>→</mo><mi>sF</mi><mo>→</mo><mi>sF</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> sSets \to sF \to sF \to Top </annotation></semantics></math></div> <p>To gain some intuition, consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mo>.</mo></msub><mo>=</mo><msub><mi>Δ</mi> <mi>n</mi></msub><mo>=</mo><mi>Hom</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y_.=\Delta_n=Hom(-,[n])</annotation></semantics></math> the standard simplex. Then by Remark 2.4.1-2 of <a href="http://www.math.uiuc.edu/~dan/Courses/2003/Spring/416/GraysonKtheory.pdf">Grayson</a> the standard geometric simplex is the set of monotone functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0.1</mn><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0.1]\to [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>Hom</mi> <mi>preorders</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msub><mo stretchy="false">]</mo> <mo>≤</mo></msub><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>=</mo><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>Hom</mi> <mi>preorders</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msub><mo stretchy="false">]</mo> <mo>≤</mo></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>Δ</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{preorders}([0,1]_\leq), [n] ) = Hom ( Hom_{preorders}(-, [0,1]_\leq), \Delta_n )</annotation></semantics></math></div> <p>equipped with a metric</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">dist</mo><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mi>inf</mi><mo stretchy="false">{</mo><mi>ϵ</mi><mo>:</mo><mo>∀</mo><mi>x</mi><mo>∃</mo><mi>y</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∧</mo><mo>∀</mo><mi>y</mi><mo>∃</mo><mi>x</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\dist(f,g)=\inf \{ \epsilon: \forall x \exists y ( f(x)=f(y) ) \wedge \forall y \exists x ( f(x)=g(y) ) \}</annotation></semantics></math></div> <p>reminiscent of Skorokhod metric in probability theory. Now instead of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_n</annotation></semantics></math> take an arbitrary simplicial set, and rewrite the definition of Skorokhod convergence in terms of the notion of smallness in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sF</mi></mrow><annotation encoding="application/x-tex">sF</annotation></semantics></math>.</p> <h3 id="geometric_realization_preserves_fibrations">Geometric realization preserves fibrations</h3> <p> <div class='num_theorem'> <h6>Theorem</h6> <p>The geometric realization of a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a> is a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>.</p> </div> <div class='proof'> <h6>Proof</h6> <p>This is shown in <a href="#Quillen68">Quillen 68</a>.</p> </div> </p> <p>This result implies that the geometric realization functor preserves all five classes of maps in a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>: <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a>, <a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a>, <a class="existingWikiWord" href="/nlab/show/acyclic+cofibrations">acyclic cofibrations</a>, <a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a>, and <a class="existingWikiWord" href="/nlab/show/acyclic+fibrations">acyclic fibrations</a>.</p> <p>In fact the geometric realization of a Kan fibration is even a <a class="existingWikiWord" href="/nlab/show/Hurewicz+fibration">Hurewicz fibration</a> (at least relative to a <a class="existingWikiWord" href="/nlab/show/convenient+category+of+spaces">convenient category of spaces</a> in which it lives). This follows from the fact that <a class="existingWikiWord" href="/nlab/show/a+Serre+fibration+between+CW-complexes+is+a+Hurewicz+fibration">a Serre fibration between CW-complexes is a Hurewicz fibration</a>; a direct proof along the lines of Quillen’s can be found in <a href="#FP90">Fritch and Piccinini, Theorem 4.5.25</a>.</p> <h3 id="induced_properties_of_the_fibrant_replacement">Induced properties of the fibrant replacement</h3> <p>The previous two sections show that the geometric realization preserves finite limits and fibrations. Since its right adjoint, the singular complex functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">Top \to sSet</annotation></semantics></math>, also preserves both (much more trivially), and since all objects of <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> are fibrant and the adjunction is simplicially enriched, it follows that the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi><mo>→</mo><mi>Top</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet \to Top \to sSet</annotation></semantics></math> is a simplicially enriched <a class="existingWikiWord" href="/nlab/show/fibrant+replacement+functor">fibrant replacement functor</a> 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> that additionally preserves both finite limits and fibrations.</p> <h3 id="geometric_realization_of_barycentric_subdivisions">Geometric realization of barycentric subdivisions</h3> <p> <div class='num_theorem'> <h6>Theorem</h6> <p>(<a href="#FP1967">Fritsch–Puppe, 1967</a>; <a href="#F1974">Fritsch 1974</a>.) There is a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</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>Sd</mi><mi>X</mi><mo stretchy="false">|</mo><mo>→</mo><mo stretchy="false">|</mo><mi>X</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|Sd X| \to |X|</annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> of the <a class="existingWikiWord" href="/nlab/show/barycentric+subdivision">barycentric subdivision</a> of 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>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the geometric realization 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 homeomorphism is homotopic to the geometric realization of the last vertex map. The homeomorphism turns the <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>Sd</mi><mi>X</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|Sd X|</annotation></semantics></math> into a subdivision of the <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>X</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|X|</annotation></semantics></math>. The statement also holds relative a simplicial subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A\subset X</annotation></semantics></math>.</p> </div> </p> <p>For an expository account, see <a href="#FP1990">Fritsch–Piccinini</a>.</p> <h2 id="examples">Examples</h2> <ul> <li>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/group">group</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>, its one-object <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> obtained by <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(\mathbf{B}G)</annotation></semantics></math> the corresponding simplicial <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>, we have that the geometric realization<div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mi>G</mi><mo>=</mo><mo stretchy="false">|</mo><mi>N</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex"> \mathcal{B}G = |N\mathbf{B}G| </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> that is the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>s (<a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a>s), as long as we give <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> the <a class="existingWikiWord" href="/nlab/show/discrete+topology">discrete topology</a>.</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><strong>geometric realization</strong></p> <ul> <li>of <a class="existingWikiWord" href="/nlab/show/geometric+realization+of+categories">categories</a>, <a class="existingWikiWord" href="/nlab/show/geometric+realization+of+simplicial+topological+spaces">simplicial topological spaces</a>, <a class="existingWikiWord" href="/nlab/show/geometric+realization+of+cohesive+%E2%88%9E-groupoids">cohesive ∞-groupoids</a>, <a class="existingWikiWord" href="/nlab/show/geometric+realisation+of+cubical+sets">cubical sets</a>.</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totalization">totalization</a></p> </li> <li> <p><span class="newWikiWord">singular complex functor<a href="/nlab/new/singular+complex+functor">?</a></span></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>Original articles:</p> <ul> <li id="Quillen68"><a class="existingWikiWord" href="/nlab/show/Daniel+Quillen">Daniel Quillen</a>, <em>The geometric realization of a Kan fibration is a Serre fibration. Proc. Amer. Math. Soc. 19 1968 1499–1500. <a href="https://www.ams.org/journals/proc/1968-019-06/S0002-9939-1968-0238322-1/S0002-9939-1968-0238322-1.pdf">pdf</a></em></li> </ul> <p>Monographs:</p> <ul> <li id="GoerssJardine09"> <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 VII.3 of: <em><a class="existingWikiWord" href="/nlab/show/Simplicial+homotopy+theory">Simplicial homotopy theory</a></em>, Progress in Mathematics, Birkhäuser (1999)</p> <p>Modern Birkh[er 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>]</p> </li> <li id="FP90"> <p><a class="existingWikiWord" href="/nlab/show/Rudolf+Fritsch">Rudolf Fritsch</a>, <a class="existingWikiWord" href="/nlab/show/Renzo+A.+Piccinini">Renzo A. Piccinini</a> §4.2 and §4.3 in: <em>Cellular structures in topology</em>, Cambridge studies in advanced mathematics <strong>19</strong> Cambridge University Press (1990) [<a href="https://doi.org/10.1017/CBO9780511983948">doi:10.1017/CBO9780511983948</a>, <a href="https://epub.ub.uni-muenchen.de/4493/1/4493.pdf">pdf</a>]</p> </li> </ul> <p>Lecture notes:</p> <ul> <li id="Lackenby08"><a class="existingWikiWord" href="/nlab/show/Marc+Lackenby">Marc Lackenby</a>, Section I.2 of: <em>Topology and Groups</em> (2008, 2018) [<a href="http://people.maths.ox.ac.uk/lackenby/tg050908.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Lackenby-TopologyAndGroups.pdf" title="pdf">pdf</a>]</li> </ul> <p>Generalization to realization of <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a>:</p> <ul> <li id="Simpson96"><a class="existingWikiWord" href="/nlab/show/Carlos+Simpson">Carlos Simpson</a>, <em>The topological realization of a simplicial presheaf</em> [<a href="https://arxiv.org/abs/q-alg/9609004">arXiv:q-alg/9609004</a>]</li> </ul> <h3 id="compatibility_with_homotopy_limits">Compatibility with homotopy limits</h3> <p>Discussion of sufficient conditions for homotopy geometric realization to be compatible with <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> (see also at <em><a class="existingWikiWord" href="/nlab/show/geometric+realization+of+simplicial+topological+spaces">geometric realization of simplicial topological spaces</a></em>):</p> <ul> <li> <p>D. Anderson, <em>Fibrations and geometric realization</em> , Bull. Amer. Math. Soc. Volume 84, Number 5 (1978), 765-788. (<a href="http://projecteuclid.org/euclid.bams/1183541139">euclid:1183541139</a>)</p> </li> <li id="Rezk14"> <p><a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <em>When are homotopy colimits compatible with homotopy base change?</em>, 2014 (<a href="https://faculty.math.illinois.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/RezkHomotopyColimitsBaseChange.pdf" title="pdf">pdf</a>)</p> </li> <li> <p>Edoardo Lanari, <em>Compatibility of homotopy colimits and homotopy pullbacks of simplicial presheaves</em> (<a href="http://algant.eu/documents/theses/lanari.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/LanariHomotopyColimitsBaseChange.pdf" title="pdf">pdf</a>)</p> <p>(expanded version of <a href="#Rezk14">Rezk 14</a>)</p> </li> </ul> <h3 id="geometric_realization_of_barycentric_subdivisions_2">Geometric realization of barycentric subdivisions</h3> <ul> <li id="FP1967"> <p><a class="existingWikiWord" href="/nlab/show/Rudolf+Fritsch">Rudolf Fritsch</a>, <a class="existingWikiWord" href="/nlab/show/Dieter+Puppe">Dieter Puppe</a>, <em>Die Homöomorphie der geometrischen Realisierungen einer semisimplizialen Menge und ihrer Normalunterteilung</em>, Arch. Math. (Basel) 18 (1967), 508–512.</p> </li> <li id="F1974"> <p>Relative semisimpliziale Approximation, Arch. Math. (Basel) 25 (1974), 75–78.</p> </li> <li id="FP1990"> <p><a class="existingWikiWord" href="/nlab/show/Rudolf+Fritsch">Rudolf Fritsch</a>, <a class="existingWikiWord" href="/nlab/show/Renzo+A.+Piccinini">Renzo A. Piccinini</a>, <em>Cellular structures in topology</em>, CUP, 1990.</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 5, 2024 at 08:25:39. 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