nLab nonabelian sheaf cohomology

Nonabelian sheaf cohomology is a notion in the context of nonabelian cohomology:

Given an infinity-category-valued (pseudo)presheaf $\mathbf{A} : Spaces^{op} \to \infty-Cat$, its cohomology $H(-,\mathbf{A})$ is the $\infty$-category valued (pseudo)presheaf which assigns to each space the directed limit over all descent data:

$$H(X, \mathbf{A}) := colim_{Y^\bullet \to X} Desc(Y^\bullet, \mathbf{A}) \,.$$

Here the colimit is over all hypercovers of $X$.

Remarks

• Dual to nonabelian sheaf cohomology is nonabelian cosheaf homotopy.

• A presheaf whose value on each space is equivalent to its descent $\infty$-category for any cover of that space is an infinity-stack.