Cohomology of categories and extensions of twisted diagrams of groups (Q659884)

If \(G\) and \(N\) are groups and let \(G\rightarrow Out(N)\) be a group homomorphism. The question if this morphism comes from an extension of \(G\) by \(N\) is decided by the vanishing of an element in \(H^3(G,Z(N))\). The equivalence classes of extensions are parameterised by \(H^2(G,Z(N))\). In the paper under review the author defines \({\mathcal R}ep\), the category whose objects are groups and morphisms from \(G\) to \(H\) are group homomorphisms from \(G\) to \(H\) modulo inner automorphisms of \(H\). The author develops an analogous theory replacing groups by functors from an indexing category \({\mathcal C}\) to \({\mathcal R}ep\).