Let G and H be two groups. There is a one-to-one correspondence between:


(i) an (isomorphism class of) extension of G by H, i.e. an exact sequence of group morphisms $1\to H\to E\to G\to 1$;


(ii) an (isomorphism class of) action of the group G on the category of H-sets.


Actually, this can even be made an equivalence of categories.


This is probably well-known by category-theoretists, but I found no reference on the subject. Does anybody know of an article or book where it has been treated ?


Lots of thanks, in advance.



Group extensions and actions on categories
The beginning of SGA 7, Exposé VII, essentially proves an equivalence between extensions of $G$ by $H$ and monoidal functors from $G$ (thought of as a discrete monoidal category) to the monoidal category of $H$-bitorsors. The correspondence here takes an extension $1 \to H \to E \stackrel{\pi}{\to} G \to 1$ to the functor sending $g \in G$ to the fiber $\pi^{-1}(g)$, which naturally carries the structure of an $H$-bitorsor via the multiplication in $E$.


This latter category can be identified with the monoidal category of autoequivalences of the category of $H$-sets: an autoequivalence of the category of $H$-sets is given as tensoring by an $H$-bitorsor, and natural isomorphisms between autoequivalences correspond to isomorphisms of $H$-bitorsors. Off the top of my head, I don't know of a reference that describes this latter equivalence explicitly.


