# nLab cohomological dimension

cohomology

### Theorems

#### $\left(\infty ,1\right)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Idea

An object in an (∞,1)-topos is said to have cohomological dimension $\le n$ if all cohomology groups of degree $k>n$ vanish on that object.

## Definition

###### Definition

For $H$ an (∞,1)-topos and $n\in ℕ$ , an object $X\in H$ is said to have cohomological dimension $\le n$ if for all Eilenberg-MacLane objects ${B}^{k}A$ for $k>n$ the cohomology of $X$ with these coefficients vanishes:

${H}^{k}\left(X,A\right):={\pi }_{0}H\left(X,{B}^{A}\right)\simeq *\phantom{\rule{thinmathspace}{0ex}}.$H^k(X, A) := \pi_0 \mathbf{H}(X,\mathbf{B}^ A) \simeq * \,.

We say the (∞,1)-topos $H$ itself has cohomological dimension $\le n$ if its terminal object does.

This appears as HTT, def. 7.2.2.18.

## Properties

###### Proposition

If $𝒳$ has homotopy dimension $\le n$ then it also has cohomology dimension $\le n$.

The converse holds if $𝒳$ has finite homotopy dimension an $n\ge 2$.

This appears as HTT, cor. 7.2.2.30.

## References

The general $\left(\infty ,1\right)$-topos-theoretic notion is discussed in section 7.2.2 of

Revised on December 6, 2011 15:19:40 by Stephan Alexander Spahn (79.227.144.188)