nLab
nonabelian sheaf cohomology

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

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

$$H(X,A):={\mathrm{colim}}_{Y^{•}\to X}\mathrm{Desc}(Y^{•},A).$$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 $\mathrm{\infty }$-category for any cover of that space is an infinity-stack.