# nLab nerve theorem

Contents

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.

# Contents

## Idea

The nerve theorem asserts that the homotopy type of a sufficiently nice topological space is encoded in the Cech nerve of a good cover.

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

###### 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 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 summand space by the point. Let $|\tilde C(\{U_i\})|$ be the geometric realization.

This is homotopy equivalent to $X$.

This is usually attributed to (Borsuk1948). The proof relies on the existence of partitions of unity (see for instance the review Hatcher, prop. 4G.2).

###### 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 Euclidean-topological ∞-groupoid : Geometric homotopy for details.

Some earlier references on the nerve theorem include

• K. Borsuk, On the imbedding of systems of compacta in simplicial complexes , Fund. Math. 35, (1948) 217–234
• 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.

• André Weil, Sur les theoremes de de Rham, Comment. Math. Helv. 26 (1952), 119–145.

(See §6.)

• Michael C. McCord, Homotopy type comparison of a space with complexes associated with its open covers, Proc. Amer. Math. Soc. 18 (1967), 705–708.

• Graeme Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105–112. (See §4.)

• Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436–491. (See Theorem 8.2.1.)

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

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)

A nerve theorem for categories has been proved in

• Kohei Tanaka, Cech complexes for covers of small categories, Homology, Homotopy and Applications 19(1), (2017), pp. 281-291. ArXiv Abstract 1508.03688