Question : How does Crossed modules comes into the set up of gerbes.

I am reading notes on 1- and 2-gerbes by Lawrence Breen. Once he defines torsors, he introduces notion of crossed modules. It was not clear how crossed modules come into picture of gerbes suddenly from nowhere.

Some search on internet gives an expository article which gives a proof of result that

Third cohomology group $H^3(G, M)$ of a group $G$ with coefficients in an abelian $G$-module $M$ is in bijection to the set $Ext^2(G, M)$ of equivalence classes of crossed module extensions of $G$ with $M$.

This gives some vague reason why crossed modules comes into the study of gerbes.

As gerbes are seen as elements of third cohomology class here, I am expecting that understanding about crossed modules in same setup as that of gerbes might give better idea of gerbes.

Can some one please tell me if this one of the reasons why crossed modules comes into picture when studying gerbes? Is there similar result as what I have quoted above in case of sheaf/Cech cohomology where gerbes are studied.

Any references will be useful.

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    $\begingroup$ Gerbes can be considered as sheaves of 2-groupoids which locally are non-empty and connected. Crossed-modules are algebraic models for pointed connected 2-groupoids (see, e.g., arxiv.org/abs/math/0512106). The relation is hence rather direct. $\endgroup$ – Yonatan Harpaz May 9 '18 at 20:15
  • $\begingroup$ I believe This definitely gives some idea but reading 29 pages might take some time. If you are free, is it possible to write as an answer the crux of that article. Please, @YonatanHarpaz $\endgroup$ – Praphulla Koushik May 10 '18 at 8:48
  • $\begingroup$ I have a very different answer to your question, but I'm not sure it is what you're looking for. For me, and some others, Gerbes are the template, i.e. a generalization of principal bundles, on which surface-holonomy is studied. IF you think of Gerbes in this way, then using crossed modules gives us the desired "commutativity" (in the form of an interchange-law) one would want when studying surface-holonomy. This answer, at least for me, have very little to do with a cohomology-inspired answer. $\endgroup$ – cheyne Jan 8 at 16:43
  • $\begingroup$ I will be very happy to see your answer.. if you think it is not suitable for this question, I can ask separate question or we can open a chat room or I can email you.. anything is fine @cheyne $\endgroup$ – Praphulla Koushik Jan 9 at 2:19
  • 1
    $\begingroup$ @cheyne koushik16@iisertvm.ac.in $\endgroup$ – Praphulla Koushik Jan 9 at 3:33

This answer is meant to elaborate on my remark above concerning crossed modules as models for 2-groupoids. A crossed module consists of a group $G$ acting on a group $H$, together with a $G$-equivariant group homomorphism $f:H \to G$ (where we consider $G$ as acting on itself by conjugation). From this data we can construct a 2-groupoid $X$ as follows: $X$ has a single object $x_0 \in X$, the $1$-morphisms from $x_0$ to itself are the elements of $G$ (with composition given by the group structure of $G$), and for $g_0,g_1 \in G$, the $2$-morphisms from $g_0$ to $g_1$ are the elements $h \in H$ such that $f(h)g_0 = g_1$. Vertical composition of 2-morphisms is then given by the group structure of $H$, while the vertical composition uses both this and the action of $G$ on $H$. More precisely, if $h \in H$ is such that $f(h)g_0 = g_1$ and $h' \in H$ is such that $f(h')g_0'= g_1'$ then their horizontal composition is the element $h(h')^{g_0} \in H$ (where $(h')^{g_0}$ denotes the action of $g_0$ on $h'$), considered as a 2-morphism from $g_0g'_0$ to $g_1g_1'$.

It can then be shown that all non-empty connected 2-groupoids arise in this way, i.e., are equivalent (in the sense of 2-groupoids) to a 2-groupoid that comes from a crossed module. This is not completely obvious: in principle, given a connected 2-groupoid $X$ and an object $x_0 \in X$ we may be inclined to construct the corresponding crossed module by setting $G = {\rm Hom}_X(x_0,x_0)$ and letting $H$ be the set of pairs $(\alpha,h)$ where $\alpha: x_0 \to x_0$ is a 1-morphism from $x_0$ to itself and $h$ is a 2-morphism from the identify on $x_0$ to $\alpha$. This construction is in principle "correct", only that it may give $G$ and $H$ which are not groups but only monoids: indeed, $X$ being a 2-groupoid doesn't mean that every 1-morphism is invertible on the nose, only that it is invertible up to an invertible 2-morphism. Nonetheless, one can still show that there is a way to choose $G$ and $H$ such that the resulting 2-groupoid will be equivalent to $X$. There is of course also the issue that in order to reconstruct the cross module we had to choose a base point in $X$. This reflects the fact that cross modules don't model 2-groupoids, but rather pointed 2-groupoids, i.e., 2-groupoids equipped with a base point (also known as 2-groups).

  • $\begingroup$ Thanks, this is definitely helps. :) I will need some time to get some idea from this. link.springer.com/chapter/10.1007/978-0-8176-4731-5_5 says how gerbes are sheaves of groupoids but they consider only abelian gerbes. $\endgroup$ – Praphulla Koushik May 13 '18 at 3:20
  • $\begingroup$ I have given bounty but not accepted the answer as i don’t completely understand, which does not mean that this is not sufficient. I will ask if I have any doubts, once I have understand completely I will accept it, thank you,. $\endgroup$ – Praphulla Koushik May 14 '18 at 3:57

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