nLab
nonabelian groupoid cohomology

Idea

Nonabelian groupoid cohomology is the nonabelian cohomology of groupoids.

This generalizes nonabelian group cohomology, which is the special case of nonabelian group cohomology for groupoids that are deloopings $BG$ of groups $G$.

Examples

Atiyah Lie groupoid

The Atiyah Lie groupoid $\mathrm{At}(P)\to \Pi (X)$ of a principal bundle $P\to X$ is the extension of $\Pi (X)$ classified by the nonabelian groupoid cocycle $\Pi (X)\to B\mathrm{AUT}(G)$ coming from a connection on $P$.