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12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> fundamental theorem of covering spaces </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/13742/#Item_8" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" 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href="/nlab/show/differential+topology">differential topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></em> and <em><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></em></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology">Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>, <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+for+the+topology">base for the topology</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood+base">neighbourhood base</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finer+topology">finer/coarser topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a>, <a class="existingWikiWord" href="/nlab/show/topological+interior">interior</a>, <a class="existingWikiWord" href="/nlab/show/topological+boundary">boundary</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separation+axiom">separation</a>, <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sobriety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/uniformly+continuous+function">uniformly continuous function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+embedding">embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+map">open map</a>, <a class="existingWikiWord" href="/nlab/show/closed+map">closed map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a>, <a class="existingWikiWord" href="/nlab/show/sub-net">sub-net</a>, <a class="existingWikiWord" href="/nlab/show/filter">filter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/convergence">convergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a><a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href="Top#UniversalConstructions">Universal constructions</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a>, <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>,</p> </li> <li> <p>fiber space, <a class="existingWikiWord" href="/nlab/show/space+attachment">space attachment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product+space">product space</a>, <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+telescope">mapping telescope</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colimits+of+normal+spaces">colimits of normal spaces</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>, <a class="existingWikiWord" href="/nlab/show/metrisable+space">metrisable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+space">Kolmogorov space</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, <a class="existingWikiWord" href="/nlab/show/regular+space">regular space</a>, <a class="existingWikiWord" href="/nlab/show/normal+space">normal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sober+space">sober space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>, <a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a></p> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+topological+space">sequentially compact</a>, <a class="existingWikiWord" href="/nlab/show/countably+compact+topological+space">countably compact</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>, <a class="existingWikiWord" href="/nlab/show/sigma-compact+topological+space">sigma-compact</a>, <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a>, <a class="existingWikiWord" href="/nlab/show/countably+paracompact+topological+space">countably paracompact</a>, <a class="existingWikiWord" href="/nlab/show/strongly+compact+topological+space">strongly compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+space">second-countable space</a>, <a class="existingWikiWord" href="/nlab/show/first-countable+space">first-countable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a>, <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+space">connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+connected+space">locally connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simply-connected+space">simply-connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+simply-connected+space">locally simply-connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a>, <a class="existingWikiWord" href="/nlab/show/point+space">point space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+space">codiscrete space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/order+topology">order topology</a>, <a class="existingWikiWord" href="/nlab/show/specialization+topology">specialization topology</a>, <a class="existingWikiWord" href="/nlab/show/Scott+topology">Scott topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, <a class="existingWikiWord" href="/nlab/show/plane">plane</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>, <a class="existingWikiWord" href="/nlab/show/cone">cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, <a class="existingWikiWord" href="/nlab/show/ball">ball</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle">circle</a>, <a class="existingWikiWord" href="/nlab/show/torus">torus</a>, <a class="existingWikiWord" href="/nlab/show/annulus">annulus</a>, <a class="existingWikiWord" href="/nlab/show/Moebius+strip">Moebius strip</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polytope">polytope</a>, <a class="existingWikiWord" href="/nlab/show/polyhedron">polyhedron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+space">projective space</a> (<a class="existingWikiWord" href="/nlab/show/real+projective+space">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configuration space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path">path</a>, <a class="existingWikiWord" href="/nlab/show/loop">loop</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>: <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>, <a class="existingWikiWord" href="/nlab/show/topology+of+uniform+convergence">topology of uniform convergence</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/path+space">path space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cantor+space">Cantor space</a>, <a class="existingWikiWord" href="/nlab/show/Mandelbrot+space">Mandelbrot space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+curve">Peano curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+with+two+origins">line with two origins</a>, <a class="existingWikiWord" href="/nlab/show/long+line">long line</a>, <a class="existingWikiWord" href="/nlab/show/Sorgenfrey+line">Sorgenfrey line</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-topology">K-topology</a>, <a class="existingWikiWord" href="/nlab/show/Dowker+space">Dowker space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Warsaw+circle">Warsaw circle</a>, <a class="existingWikiWord" href="/nlab/show/Hawaiian+earring+space">Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces+are+sober">Hausdorff spaces are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces">closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact">open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff">quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lebesgue+number+lemma">Lebesgue number lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces">sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+totally+bounded">sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous">continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+are+normal">paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity">paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+injections+are+embeddings">closed injections are embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+maps+to+locally+compact+spaces+are+closed">proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings">injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact">locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+second-countable+spaces+are+sigma-compact">locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+regular+spaces+are+paracompact">second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CW-complexes+are+paracompact+Hausdorff+spaces">CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urysohn%27s+lemma">Urysohn's lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tychonoff+theorem">Tychonoff theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tube+lemma">tube lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael%27s+theorem">Michael's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brouwer%27s+fixed+point+theorem">Brouwer's fixed point theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jordan+curve+theorem">Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+cofiber+sequence">cofiber sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+category">Strøm model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></p> </li> </ul> </div></div> <h4 id="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> <h4 id="bundles">Bundles</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/bundles">bundles</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/parameterized+stable+homotopy+theory">stable</a>) <a class="existingWikiWord" href="/nlab/show/parameterized+homotopy+theory">parameterized homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> </ul> <h2 id="sidebar_context">Context</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/slice+topos">slice topos</a>, <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/dependent+linear+type+theory">linear</a>) <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a></p> </li> </ul> <h2 id="sidebar_classes_of_bundles">Classes of bundles</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/retractive+space">retractive space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a>, <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/numerable+bundle">numerable bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+bundle">sphere bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+bundle">projective bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+3-bundle">principal 3-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+bundle">circle bundle</a>, <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation+bundle">orientation bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a>, <a class="existingWikiWord" href="/nlab/show/stringor+bundle">stringor bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a>, <a class="existingWikiWord" href="/nlab/show/2-gerbe">2-gerbe</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-gerbe">∞-gerbe</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+coefficient+bundle">local coefficient bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/2-vector+bundle">2-vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/real+vector+bundle">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+vector+bundle">complex</a>/<a class="existingWikiWord" href="/nlab/show/holomorphic+vector+bundle">holomorphic</a>, <a class="existingWikiWord" href="/nlab/show/quaternionic+vector+bundle">quaternionic</a></p> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological</a>, <a class="existingWikiWord" href="/nlab/show/differentiable+vector+bundle">differentiable</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+vector+bundle">algebraic</a></p> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+vector+bundle">with connection</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/complex+line+bundle">complex</a>, <a class="existingWikiWord" href="/nlab/show/holomorphic+line+bundle">holomorphic</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+line+bundle">algebraic</a></p> <p><a class="existingWikiWord" href="/nlab/show/cubical+structure+on+a+line+bundle">cubical structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a><a class="existingWikiWord" href="/nlab/show/Vect%28X%29">of vector bundles</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/VectBund">VectBund</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum</a>, <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+bundles">tensor product</a>, <a class="existingWikiWord" href="/nlab/show/external+tensor+product+of+vector+bundles">external tensor product</a>, <a class="existingWikiWord" href="/nlab/show/inner+product+of+vector+bundles">inner product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+vector+bundle">dual vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+vector+bundle">stable vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/virtual+vector+bundle">virtual vector bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+of+spectra">bundle of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+bundle">natural bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+bundle">equivariant bundle</a></p> </li> </ul> <h2 id="sidebar_universal_bundles">Universal bundles</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+vector+bundle">universal vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/universal+complex+line+bundle">universal complex line bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a>, <a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></p> </li> </ul> <h2 id="sidebar_presentations">Presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microbundle">microbundle</a></p> </li> </ul> <h2 id="sidebar_examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+bundle">empty bundle</a>, <a class="existingWikiWord" href="/nlab/show/zero+bundle">zero bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tautological+line+bundle">tautological line bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/basic+line+bundle+on+the+2-sphere">basic line bundle on the 2-sphere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+fibration">Hopf fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+line+bundle">canonical line bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+circle+bundle">prequantum circle bundle</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a></p> </li> </ul> <h2 id="sidebar_constructions">Constructions</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/clutching+construction">clutching construction</a></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='#statement'>Statement</a></li> <li><a href='#applications'>Applications</a></li> <li><a href='#in_homotopy_type_theory'>In homotopy type theory</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a>, the <em>fundamental theorem of covering spaces</em> says that for a sufficiently well-behaved <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/functor">functor</a> which sends a <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</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> to its <a class="existingWikiWord" href="/nlab/show/monodromy">monodromy</a> <a class="existingWikiWord" href="/nlab/show/Set">Set</a>-<a class="existingWikiWord" href="/nlab/show/action">action</a> (<a class="existingWikiWord" href="/nlab/show/permutation+representation">permutation representation</a>) of the <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</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 the <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a>.</p> <p id="PathLifting"> This is a basic instance of the general principle of <a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a>.</p> <p><img src="https://ncatlab.org/nlab/files/PathLiftingInCoveringSpace-230214.jpg" width="500" /></p> <p>It follows in particular that for <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected</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> the <a class="existingWikiWord" href="/nlab/show/automorphism+group">automorphism group</a> of the <a class="existingWikiWord" href="/nlab/show/universal+covering+space">universal covering space</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> coincides with the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X,x)</annotation></semantics></math> itself (for any basepoint <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 often yields a convenient means to determine the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</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> in the first place.</p> <p>This is closely related to the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a>; the equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><msub><mi>Grpd</mi> <mrow><mo stretchy="false">/</mo><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mi>X</mi></mrow></msub><mo>≃</mo><mn>∞</mn><msup><mi>Grpd</mi> <mrow><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mi>X</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\infty Grpd_{/\esh X} \simeq \infty Grpd^{\esh X}</annotation></semantics></math> restricts to an equivalence between the subcategories of bundles <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y \to \esh X</annotation></semantics></math> with 0-truncated fibers and of set-valued functors on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\esh X</annotation></semantics></math>. Observe in particular that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msup><mo>≃</mo><msup><mi>Set</mi> <mrow><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mi>X</mi></mrow></msup></mrow><annotation encoding="application/x-tex">Set^{\Pi_1(X)} \simeq Set^{\esh X}</annotation></semantics></math>.</p> <p>(Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\esh X</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/shape+modality">shape</a> of the topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, <a class="existingWikiWord" href="/nlab/show/shape+via+cohesive+path+%E2%88%9E-groupoid">hence</a> its <a class="existingWikiWord" href="/nlab/show/fundamental+infinity-groupoid">fundamental <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>-groupoid</a>).</p> <h2 id="statement">Statement</h2> <div class="num_theorem" id="FundamentalTheoremOfCoveringSpaces"> <h6 id="theorem">Theorem</h6> <p><strong>(fundamental theorem of covering spaces)</strong></p> <p>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> be a <a class="existingWikiWord" href="/nlab/show/locally+path-connected+topological+space">locally path-connected</a> and <a class="existingWikiWord" href="/nlab/show/semi-locally+simply-connected+topological+space">semi-locally simply-connected topological space</a>. Then the operations on</p> <ol> <li> <p>extracting the <a class="existingWikiWord" href="/nlab/show/monodromy">monodromy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Fib</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">Fib_{E}</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reconstructing+a+covering+space+from+monodromy">reconstructing a covering space from monodromy</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mi>Rec</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\; Rec(\rho)</annotation></semantics></math></p> </li> </ol> <p>constitute an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Cov</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><munderover><mo>≃</mo><munder><mo>⟶</mo><mi>Fib</mi></munder><mover><mo>⟵</mo><mi>Rec</mi></mover></munderover><msup><mi>Set</mi> <mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> Cov(X) \underoverset {\underset{Fib}{\longrightarrow}} {\overset{Rec}{\longleftarrow}} {\simeq} Set^{\Pi_1(X)} </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/category+of+covering+spaces">category of covering spaces</a>, and the category of <a class="existingWikiWord" href="/nlab/show/permutation+representation">permutation</a> <a class="existingWikiWord" href="/nlab/show/groupoid+representations">groupoid representations</a> of the <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</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>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>With the standard definitions of the two functors, both are in fact inverse <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> of categories instead of just <a class="existingWikiWord" href="/nlab/show/equivalences+of+categories">equivalences of categories</a> (meaning that the required <a class="existingWikiWord" href="/nlab/show/natural+isomorphisms">natural isomorphisms</a> from the composites of the two functors to the <a class="existingWikiWord" href="/nlab/show/identity+functor">identity functor</a> are componentwise <a class="existingWikiWord" href="/nlab/show/equalities">equalities</a>), which establishes the claim right away. For definiteness, we make this explicit:</p> <p>Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>∈</mo><msup><mi>Set</mi> <mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\rho \in Set^{\Pi_1(X)}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/permutation+representation">permutation representation</a>, we need to exhibit a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> of permutation representations.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>ρ</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>ρ</mi><mo>⟶</mo><mi>Fib</mi><mo stretchy="false">(</mo><mi>Rec</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \eta_{\rho} \;\colon\; \rho \longrightarrow Fib(Rec(\rho)) </annotation></semantics></math></div> <p>First consider what the right hand side is like: By <a href="reconstruction+of+covering+spaces+from+monodromy#ElementaryReconstructionCoveringSpace">this def.</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Rec</mi></mrow><annotation encoding="application/x-tex">Rec</annotation></semantics></math> and <a href="monodromy#CoveringSpaceMonodromy">this def.</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fib</mi></mrow><annotation encoding="application/x-tex">Fib</annotation></semantics></math> we have for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> an actual equality</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Fib</mi><mo stretchy="false">(</mo><mi>Rec</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Fib(Rec(\rho))(x) = \rho(x) \,. </annotation></semantics></math></div> <p>To similarly understand the value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fib</mi><mo stretchy="false">(</mo><mi>Rec</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Fib(Rec(\rho))</annotation></semantics></math> on morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>γ</mi><mo stretchy="false">]</mo><mo>∈</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[\gamma] \in \Pi_1(X)</annotation></semantics></math>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\gamma \colon [0,1] \to X</annotation></semantics></math> be a representing <a class="existingWikiWord" href="/nlab/show/path">path</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 in the proof of the path lifting lemma for covering spaces (<a href="covering+space#CoveringSpacePathLifting">this lemma</a>) we find a <a class="existingWikiWord" href="/nlab/show/finite+number">finite number</a> of paths <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>γ</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\{\gamma_i\}_{i \in \{1,n\}}</annotation></semantics></math> such that</p> <ol> <li> <p>regarded as morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>γ</mi> <mi>i</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\gamma_i]</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(X)</annotation></semantics></math> they <a class="existingWikiWord" href="/nlab/show/composition">compose</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>γ</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\gamma]</annotation></semantics></math>:</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>γ</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><msub><mi>γ</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo>∘</mo><mi>⋯</mi><mo>∘</mo><mo stretchy="false">[</mo><msub><mi>γ</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>∘</mo><mo stretchy="false">[</mo><msub><mi>γ</mi> <mn>1</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [\gamma] = [\gamma_n] \circ \cdots \circ [\gamma_2] \circ [\gamma_1] </annotation></semantics></math></div></li> <li> <p>each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\gamma_i</annotation></semantics></math> factors through an open subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U_i \subset X</annotation></semantics></math> over which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Rec</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Rec(\rho)</annotation></semantics></math> trivializes.</p> </li> </ol> <p>Hence by <a class="existingWikiWord" href="/nlab/show/functor">functoriality</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fib</mi><mo stretchy="false">(</mo><mi>Rec</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Fib(Rec(\rho))</annotation></semantics></math> it is sufficient to understand its value on these paths <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\gamma_i</annotation></semantics></math>. But on these we have again by direct unwinding of the definitions that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Fib</mi><mo stretchy="false">(</mo><mi>Rec</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>γ</mi> <mi>i</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>=</mo><mi>ρ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>γ</mi> <mi>i</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Fib(Rec(\rho))([\gamma_i]) = \rho([\gamma_i]) \,. </annotation></semantics></math></div> <p>This means that if we take</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>ρ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>=</mo></mover><mi>Fib</mi><mo stretchy="false">(</mo><mi>Rec</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \eta_\rho(x) \colon \rho(x) \overset{=}{\longrightarrow} Fib(Rec(\rho)) </annotation></semantics></math></div> <p>to be the above identification, then this is a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> and hence in a particular a natural isomorphism, as required.</p> <p>It remains to see that these morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>ρ</mi></msub></mrow><annotation encoding="application/x-tex">\eta_\rho</annotation></semantics></math> are themselves natural in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>, hence that for each morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>ρ</mi><mo>→</mo><mi>ρ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\phi \colon \rho \to \rho'</annotation></semantics></math> the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ρ</mi></mtd> <mtd><mover><mo>⟶</mo><mi>ϕ</mi></mover></mtd> <mtd><mi>ρ</mi><mo>′</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>eta</mi> <mi>ρ</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>η</mi> <mrow><mi>ρ</mi><mo>′</mo></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Fib</mi><mo stretchy="false">(</mo><mi>Rec</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>Fib</mi><mo stretchy="false">(</mo><mi>Rec</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>Fib</mi><mo stretchy="false">(</mo><mi>Rec</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \rho &\overset{\phi}{\longrightarrow}& \rho' \\ {}^{\mathllap{eta_\rho}}\downarrow && \downarrow^{\mathrlap{\eta_{\rho'}}} \\ Fib(Rec(\rho)) &\underset{Fib(Rec(\phi))}{\longrightarrow}& Fib(Rec(\rho')) } </annotation></semantics></math></div> <p>commutes as a diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Rep</mi><mo stretchy="false">(</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Rep(\Pi_1(X), Set)</annotation></semantics></math>. Since these morphisms are themselves groupoid homotopies (natural isomorphisms) this is the case precisely if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> the corresponding component diagram commutes. But by the above this is</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>ρ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>ρ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>=</mo></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>=</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Fib</mi><mo stretchy="false">(</mo><mi>Rec</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>Fib</mi><mo stretchy="false">(</mo><mi>Rec</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>Fib</mi><mo stretchy="false">(</mo><mi>Rec</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \rho(x) &\overset{\phi(x)}{\longrightarrow}& \rho'(x) \\ {}^{\mathllap{=}}\downarrow && \downarrow^{\mathrlap{=}} \\ Fib(Rec(\rho))(x) &\underset{Fib(Rec(\phi))(x) }{\longrightarrow}& Fib(Rec(\rho'))(x) } </annotation></semantics></math></div> <p>and hence this means that the top and bottom horizontal morphism are in fact equal. Direct unwinding of the definitions shows that this is indeed the case.</p> <p>Conversely, given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>∈</mo><mi>Cov</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E \in Cov(X)</annotation></semantics></math> a covering space, we need to exhibit a natural isomorphism of covering spaces of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ϵ</mi> <mi>E</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Rec</mi><mo stretchy="false">(</mo><mi>Fib</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⟶</mo><mi>E</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \epsilon_E \;\colon\; Rec(Fib(E)) \longrightarrow E \,. </annotation></semantics></math></div> <p>Again by <a href="reconstruction+of+covering+spaces+from+monodromy#ElementaryReconstructionCoveringSpace">this def.</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Rec</mi></mrow><annotation encoding="application/x-tex">Rec</annotation></semantics></math> and <a href="monodromy#CoveringSpaceMonodromy">this def.</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fib</mi></mrow><annotation encoding="application/x-tex">Fib</annotation></semantics></math> the underlying set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Rec</mi><mo stretchy="false">(</mo><mi>Fib</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Rec(Fib(E))</annotation></semantics></math> is actually equal to that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, hence it is sufficient to check that this <a class="existingWikiWord" href="/nlab/show/identity+function">identity function</a> on underlying sets is a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>.</p> <p>By the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/locally+path-connected+topological+space">locally path-connected</a> and <a class="existingWikiWord" href="/nlab/show/semi-locally+simply+connected+topological+space">semi-locally simply connected</a>, it is sufficient to check for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U\subset X</annotation></semantics></math> an open path-connected subset and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> a point with the property that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(U,x) \to \pi_1(X,x)</annotation></semantics></math> lands is constant on the trivial element, that the open subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><mo stretchy="false">{</mo><mover><mi>x</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">}</mo><mo>⊂</mo><msup><mi>p</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U \times \{\hat x\} \subset p^{-1}(U)</annotation></semantics></math> form a basis for the topology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Rec</mi><mo stretchy="false">(</mo><mi>Fib</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Rec(Fib(E))</annotation></semantics></math>. But this is the case by definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Rec</mi></mrow><annotation encoding="application/x-tex">Rec</annotation></semantics></math>.</p> <p>It remains to see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϵ</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">\epsilon_E</annotation></semantics></math> is itself natural in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>. But as for the converse direction, since the components of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϵ</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">\epsilon_E</annotation></semantics></math> are in fact equalities, this follows by direct unwinding of the definitions.</p> <p>This establishes an equivalence as required. In fact this is an <a class="existingWikiWord" href="/nlab/show/adjoint+equivalence">adjoint equivalence</a>.</p> </div> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <h2 id="in_homotopy_type_theory">In homotopy type theory</h2> <p>In <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>, the fundamental theorem of covering spaces is really just a special case of the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of <a class="existingWikiWord" href="/nlab/show/n-truncation+modality"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>1</mn> </mrow> <annotation encoding="application/x-tex">1</annotation> </semantics> </math>-truncation</a>.</p> <p>Indeed, in HoTT a covering space over a <a class="existingWikiWord" href="/nlab/show/type">type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is usually defined as a <a class="existingWikiWord" href="/nlab/show/dependent+type">dependent type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="sans-serif"><mi>Sets</mi></mstyle></mrow><annotation encoding="application/x-tex">C \colon X \to \mathsf{Sets}</annotation></semantics></math>. We can get the more familiar bundle definition of a covering space by taking the <a class="existingWikiWord" href="/nlab/show/dependent+sum+type">dependent sum type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi></mrow></msub><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sum_{x \colon X} C(x)</annotation></semantics></math>, which comes equipped with a canonical projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">proj</mo> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi></mrow></msub><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\operatorname{proj}_1 \colon \sum_{x \colon X} C(x) \to X</annotation></semantics></math>.</p> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>Sets</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{Sets}</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-type, then by the universal property of 1-truncation we have an equivalence:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">‖</mo><mi>X</mi><msub><mo stretchy="false">‖</mo> <mn>1</mn></msub><mo>→</mo><mstyle mathvariant="sans-serif"><mi>Sets</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="sans-serif"><mi>Sets</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \big( \Vert X \Vert_1 \to \mathsf{Sets} \big) \;\simeq\; \big( X \to \mathsf{Sets} \big) \,. </annotation></semantics></math></div> <p>Since the 1-truncation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">‖</mo><mi>X</mi><msub><mo stretchy="false">‖</mo> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\Vert X \Vert_1</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</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>, this really is the fundamental theorem of covering spaces.</p> <p><br /></p> <p id="InCohesiveHomotopyTypeTheory">Or rather, more faithful to the traditional concepts in topology, it is the <a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">ʃ</mo></mrow><annotation encoding="application/x-tex">\esh</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a> which turns a <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+type">geometric homotopy type</a> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> its <a class="existingWikiWord" href="/nlab/show/fundamental+infinity-groupoid">fundamental <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>-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\esh X</annotation></semantics></math> (see at <em><a class="existingWikiWord" href="/nlab/show/shape+via+cohesive+path+%E2%88%9E-groupoid">shape via cohesive path ∞-groupoid</a></em>) truncating to the actual <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">‖</mo><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mi>X</mi><msub><mo stretchy="false">‖</mo> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\Vert \esh X \Vert_1</annotation></semantics></math>.</p> <p>Then the <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> between the <a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> and the <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math> says that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="sans-serif"><mi>Sets</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>→</mo><mo>♭</mo><mstyle mathvariant="sans-serif"><mi>Sets</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \big( \esh X \to \mathsf{Sets} \big) \;\simeq\; \big( X \to \flat \mathsf{Sets} \big) \,, </annotation></semantics></math></div> <p>or, equivalently, that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">‖</mo><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mi>X</mi><msub><mo stretchy="false">‖</mo> <mn>1</mn></msub><mo>→</mo><mstyle mathvariant="sans-serif"><mi>Sets</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>→</mo><mo>♭</mo><mstyle mathvariant="sans-serif"><mi>Sets</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \big( \Vert \esh X \Vert_1 \to \mathsf{Sets} \big) \;\simeq\; \big( X \to \flat \mathsf{Sets} \big) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo><mi>Sets</mi></mrow><annotation encoding="application/x-tex">\flat Sets</annotation></semantics></math> is the actual classifier for covering spaces in the generality of cohesive (e.g. topological) homotopy types. This reflects the fundamental theorem of covering spaces as traditionally understood in <a class="existingWikiWord" href="/nlab/show/topology">topology</a>.</p> <p>This is the topic of <a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">dcct, sec. 3.8.6</a>, <a href="https://arxiv.org/pdf/1310.7930v1.pdf#page=358">p. 358</a>, see also <a href="#CherubiniRijke20">Cherubini & Rijke 2020, Thm. 8.7</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/transport">transport</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+cover">universal cover</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G-set">G-set</a></p> </li> <li> <p><a href="smooth+homotopy#HomotopyOfSmoothPathsRelativeToTheirEndpoints">homotopy of smooth paths relative to their endpoints</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck%27s+Galois+theory">Grothendieck's Galois theory</a></p> </li> </ul> <h2 id="references">References</h2> <p>On the classical theory:</p> <p>A detailed treatment is available in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Nicolas+Bourbaki">Nicolas Bourbaki</a>, <em>Topologie Algébrique</em>, Chapitres 1 à 4, Springer (1998, 2016) [<a href="https://doi.org/10.1007/978-3-662-49361-8">doi:10.1007/978-3-662-49361-8</a>, ISBN 978-3-662-49361-8]</li> </ul> <p>Textbook account:</p> <ul> <li id="tomDieck2008"><a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, Thm. 3.3.2 in: <em>Algebraic topology</em>, European Mathematical Society, Zürich (2008) [<a href="https://www.ems-ph.org/books/book.php?proj_nr=86">doi:10.4171/048</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/diecktop.pdf">pdf</a>]</li> </ul> <p>Lecture notes:</p> <ul> <li id="Waldhausen"> <p><a class="existingWikiWord" href="/nlab/show/Friedhelm+Waldhausen">Friedhelm Waldhausen</a>, around p. 122 of <em>Topologie</em> (<a href="https://www.math.uni-bielefeld.de/~fw/ein.pdf">pdf</a>)</p> </li> <li id="Moller11"> <p><a class="existingWikiWord" href="/nlab/show/Jesper+M%C3%B8ller">Jesper Møller</a>, Thm. 7.8 in: <em>The fundamental group and covering spaces</em> (2011) [<a href="https://arxiv.org/abs/1106.5650">arXiv:1106.5650</a>, <a href="http://www.math.ku.dk/~moller/f03/algtop/notes/covering.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Moller-FundamentalGroup.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive homotopy theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">dcct, sec. 3.8.6</a>, <a href="https://arxiv.org/pdf/1310.7930v1.pdf#page=358">p. 358</a></li> </ul> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a>:</p> <ul> <li id="CherubiniRijke20"><a class="existingWikiWord" href="/nlab/show/Felix+Cherubini">Felix Cherubini</a>, <a class="existingWikiWord" href="/nlab/show/Egbert+Rijke">Egbert Rijke</a>, Thm. 8.7 in: <em>Modal Descent</em>, Mathematical Structures in Computer Science (2020) (<a href="https://arxiv.org/abs/2003.09713">arXiv:2003.09713</a>, <a href="https://doi.org/10.1017/S0960129520000201">doi:10.1017/S0960129520000201</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 18, 2023 at 05:21:48. 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