nLab
cohomological dimension

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 $n\in \mathbb{N}$ , 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:

$$H^k(X,A):={\pi }_{0}H(X,B^A)\simeq *.$$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.

Related concepts

• dimension

  • homotopy dimension

  • cohomological dimension

  • covering dimension

  • Heyting dimension

References

The general $(\mathrm{\infty },1)$-topos-theoretic notion is discussed in section 7.2.2 of

• Jacob Lurie, Higher Topos Theory