nLab cohomological dimension

Idea

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

Definition

Definition

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

$$H^k(X, A) := \pi_0 \mathbf{H}(X,\mathbf{B}^k A) \simeq * \,.$$

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

This appears as HTT, def. 7.2.2.18.

Properties

Proposition

If $\mathcal{X}$ has homotopy dimension $\leq n$ then it also has cohomology dimension $\leq n$.

The converse holds if $\mathcal{X}$ has finite homotopy dimension and $n \geq 2$.

This appears as HTT, cor. 7.2.2.30.

Related concepts

notion of dimension

• homotopy dimension

• cohomology dimension (virtual)

• covering dimension

• Heyting dimension

• dimension of a manifold

• dimension of a cell complex

• dimension of a separable metric space

• small inductive dimension of a regular space

• large inductive dimension of a normal space

References

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

• Jacob Lurie, Higher Topos Theory