> This page is about a property of [[Cech nerves]] in [[homotopy theory]]. For the "nerve theorem" in [[category theory]] see at _[[Segal conditions]]_. For the "nerve theorem" for [[monads with arities]] see [there](monad+with+arities#NerveTheorem). *** +-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Topology +--{: .hide} [[!include topology - contents]] =-- #### Homotopy theory +--{: .hide} [[!include homotopy - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea The **nerve theorem** asserts that the [[homotopy type]] of a sufficiently nice [[topological space]] is encoded in the [Čech nerve](Čech+nerve#FromACover) of a [[good open cover]] (as used in [[Čech homology]]). This can be seen as a special case of some aspects of [[étale homotopy]] as the étale homotopy type of nice spaces coincides with the homotopy type of its Cech nerve. ## Statement +-- {: .num_theorem} ###### Theorem Let $X$ be a [[paracompact space]] and $\{U_i \to X\}$ a [[good open cover]]. Write $C(\{U_i\})$ for the [[Cech nerve]] of this cover $$ C(\{U_i\}) = \left( \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}\coprod_{i,j} U_i \cap U_j \stackrel{\longrightarrow}{\longrightarrow} \coprod_{i} U_i \right) \,, $$ (a [[simplicial object|simplicial]] [[space]]) and write $$ \tilde C(\{U_i\}) = \left( \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}\coprod_{i,j} * \stackrel{\longrightarrow}{\longrightarrow} \coprod_{i} * \right) $$ for the [[simplicial set]] obtained by replacing in $C(\{U_i\})$ each [[direct sum]]mand space by the [[point]]. Let $|\tilde C(\{U_i\})|$ be the [[geometric realization]]. This is [[homotopy equivalence|homotopy equivalent]] to $X$. =-- The proof relies on the existence of [[partitions of unity]]. This is usually attributed to ([Borsuk 1948](#Borsuk48)). It appears more explicitly as [Weil 52, p. 141](#Weil52) [McCord 67, Thm. 2](#McCord67), review in [Hatcher, prop. 4G.3](#Hatcher). +-- {: .num_remark} ###### Remark This statement implies that in the [[cohesive (∞,1)-topos]] [[ETop∞Grpd]] the intrinsic [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] coincides with the ordinary [[fundamental ∞-groupoid]] functor of [[paracompact topological spaces]]. See <a href="http://ncatlab.org/nlab/show/Euclidean-topological+infinity-groupoid#GeometricHomotopy">Euclidean-topological ∞-groupoid : Geometric homotopy</a> for details. =-- ## Related concepts * [[Čech homology]] ## References Original references: * {#Borsuk48} [[Karol Borsuk]], _On the imbedding of systems of compacta in simplicial complexes_ , Fund. Math. 35, (1948) 217–234 ([dml:213158](https://eudml.org/doc/213158)) * [[Jean Leray]], _L'anneau spectral et l'anneau filtré d'homologie d'un espace localement compact et d'une application continue_, J. Math. Pures Appl. (9) 29 (1950), 1–139. * {#Weil52} [[André Weil]], §5. in: _Sur les theoremes de de Rham_, Comment. Math. Helv. 26 (1952), 119–145 ([dml:139040](https://eudml.org/doc/139040)) * {#McCord67} [[Michael C. McCord]], _Homotopy type comparison of a space with complexes associated with its open covers_, Proc. Amer. Math. Soc. 18 (1967), 705–708 ([jstor:2035443](https://www.jstor.org/stable/2035443)) * [[Graeme Segal]], §4 in: _Classifying spaces and spectral sequences_, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105–112. * [[Armand Borel]] and [[Jean-Pierre Serre]], Theorem 8.2.1. in: _Corners and arithmetic groups_, Comment. Math. Helv. 48 (1973), 436–491. A version for hypercovers is discussed in * [[Daniel Dugger]], [[Daniel C. Isaksen]], _Topological hypercovers and $\mathbb{A}^1$-realizations_, Math. Z. 246 (2004), no. 4, 667–689 A review appears as corollary 4G.3 in the textbook * {#Hatcher} [[Allen Hatcher]], _Algebraic topology_ ([web](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html)) . Some slightly stronger statements are discussed in * Anders Björner, _Nerves, fibers and homotopy groups_, Journal of combinatorial theory, series A, 102 (2003), 88-93 * Andrzej Nagórko, _Carrier and nerve theorems in the extension theory_, Proc. Amer. Math. Soc. 135 (2007), 551-558. ([web](http://www.ams.org/journals/proc/2007-135-02/S0002-9939-06-08477-2/home.html)) A nerve theorem for categories: * Kohei Tanaka, _Cech complexes for covers of small categories_, Homology, Homotopy and Applications 19(1), (2017), pp. 281-291. [arXiv:1508.03688](https://arxiv.org/abs/1508.03688) [[!redirects nerve theorems]] [[!redirects Borsuk's nerve theorem]]