nLab
nonabelian cosheaf homotopy

Nonabelian cosheaf homotopy is a notion in the context of nonabelian cohomology:

Given an infinity-category-valued (pseudo)copresheaf $B:\mathrm{Spaces}\to \mathrm{\infty }-\mathrm{Cat}$, its homotopy $\pi (-,B)$ is the $\mathrm{\infty }$-category valued (pseudo)copresheaf which assigns to each space the limit over all codescent data:

$$\pi (X,B):={\lim }_{Y^{•}\to X}\mathrm{Codesc}(Y^{•},B).$$\pi(X, \mathbf{B}) := lim_{Y^\bullet \to X} Codesc(Y^\bullet, \mathbf{B}) \,.

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

Remarks

• Dual to nonabelian cosheaf homotopy is nonabelian sheaf cohomology.

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