# nLab nerve theorem

This page is about a property of Čech nerves in homotopy theory. For the “nerve theorem” in category theory see at Segal conditions.

### Context

#### Topology

topology

algebraic topology

# 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

$C(\{U_i\}) = \left( \stackrel{\to}{\stackrel{\to}{\to}}\coprod_{i,j} U_i \cap U_j \stackrel{\to}{\to} \coprod_{i} U_i \right) \,,$

(a simplicial space) and write

$\tilde C(\{U_i\}) = \left( \stackrel{\to}{\stackrel{\to}{\to}}\coprod_{i,j} * \stackrel{\to}{\to} \coprod_{i} * \right)$

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.

## References

The nerve theorem is usually attributed to

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

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)

Revised on July 5, 2013 17:05:36 by Urs Schreiber (89.204.137.186)