group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
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.
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:
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.
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 an $n \geq 2$.
This appears as HTT, cor. 7.2.2.30.
notion of dimension
The general $(\infty,1)$-topos-theoretic notion is discussed in section 7.2.2 of