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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.

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  • $\begingroup$ I think this question is too localized for MO. It is a basic result which is contained in every introduction to semidirect products of groups. I think that the category-theory tag is wrong. $\endgroup$
    – Martin Brandenburg
    Commented Feb 14, 2012 at 21:32
  • $\begingroup$ The category of H-sets is equivalent to the category BH with one object and H as automorphisms. Then you statement goes by the name of Schreier-Theory or Dedecker-Cocycles and dates back to the beginning of the 20th century. $\endgroup$
    – Thomas Nikolaus
    Commented Feb 14, 2012 at 21:43
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    $\begingroup$ Martin, are you are you read well the question ? This equivalence of category, if not utterly deep, is not to be found "in any introduction to semi-direct products", and is surely about category-theory. $\endgroup$
    – Joël
    Commented Feb 14, 2012 at 22:20
  • $\begingroup$ John Baez wrote about Schreier theory in his TWF series: math.ucr.edu/home/baez/week223.html -- he points out a good reference for the Schreier theory of groupoids too: arxiv.org/abs/math.CT/0410202 . $\endgroup$
    – Finn Lawler
    Commented Feb 14, 2012 at 22:38
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    $\begingroup$ (Also, I think H-Set is equivalent not to BH but to [BH, Set].) $\endgroup$
    – Finn Lawler
    Commented Feb 14, 2012 at 22:40

3 Answers 3

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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.

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  • $\begingroup$ By the way, if $A$ is an algebra, there is a similar statement for strongly $G$-graded algebras with unit component $A$, and actions of $G$ on the category of $A$-modules. Do you know if this is also done in SGA 7 ? Thanks again. $\endgroup$
    – Erwan Biland
    Commented Feb 15, 2012 at 1:40
  • $\begingroup$ Does this work also for $G$ and $H$ which are topological or Lie by using classifying their classifying topoi? $\endgroup$
    – David Carchedi
    Commented Feb 18, 2012 at 14:54
  • $\begingroup$ @Erwan: I doubt it's in SGA7, but the argument should be almost exactly the same. @David: In Grothendieck's article, he works in an arbitrary (Grothendieck) topos, so I think the answer to your question is yes (modulo checking what Grothendieck actually says, as opposed to my rough translation of it). $\endgroup$
    – Evan Jenkins
    Commented Feb 24, 2012 at 2:16
  • $\begingroup$ The equivalence between autoequivalences and bitorsors is a version of the Eilenberg-Watts theorem; it can be deduced from the universal property of the Yoneda embedding. $\endgroup$
    – Qiaochu Yuan
    Commented Aug 6, 2016 at 0:06
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Such extensions are what D. Conduché called crossed 2-modules (in French: modules croisés de longueur 2) in his 1983 paper: Modules croisés généralisés de longueurs 2, J. Pure and Appl. Alg. Vol. 34, Issues 2-3 (1984), pp. 155-178. My guess is that an action of $G$ on the category of $H$-sets amounts to what the same author called "non-abelian pre-crossed complex" (in French: complexe pré-croisé non abélien) in the same paper. The equivalence you talked about is then Théorème 2.6 of the above mentioned reference.

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Another relevant paper is

R. Brown, G. Danesh-Naruie, J.P.L. Hardy ``The fundamental groupoid as a topological groupoid'', Proc. Edinburgh Math. Soc. 19 (1975) 237-244.

Section 4 explains that an extension $1 \to B \to E \to G \to 1$ is isomorphic to an exact sequence determined by a fibration of groupoids $G \ltimes A \to G$ where $A$ is a groupoid which is a $G$-module. In the case in point, $A$ is the action groupoid determined by the action of $E$ on the right of $G$.

Understanding of this really requires the notion of covering morphism of groupoids and the equivalence of categories between covering morphisms of $G$ and actions of $G$ on sets. I feel this notion of covering morphism is somewhat neglected. However in the book

Higgins, P.J. Notes on categories and groupoids, Mathematical Studies, Volume 32. van Nostrand Reinhold Co. London (1971); Reprints in Theory and Applications of Categories, No. 7 (2005) pp 1-195.

the notion is used to prove a generalisation of Grushko's theorem. And fibrations of groupoids are also useful algebraically, and topologically!

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  • $\begingroup$ Thank you for this answer. It turns out that I might have to work with a notion of covering morphism of G-algebras, due to Puig. I do not know if the two notions are related. Anyway, your post gave me the opportunity to learn about groupoids. Thanks ! $\endgroup$
    – Erwan Biland
    Commented Feb 20, 2012 at 17:02
  • $\begingroup$ I'd be grateful for a reference to the work of Puig. There is an Exercise 8 on p. 379 of "Topology and groupoids" in which I suggested a kind of research project on dealing with a subalgebra, in fact an ideal, by related methods, but I am not sure if that is the good direction! $\endgroup$
    – Ronnie Brown
    Commented Mar 2, 2012 at 17:01

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