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here is something I’ve been fighting with lately. it should be completely n-classical in terms of 2-groups and crossed modules, but since I am sistematically lost in the 2-groups diagrammatics I’ll write here what I have in mind, in the hope someone will tell me how this construction is n-classically named :)
given an abelian group extension $1\to A\to G\to H\to 1$, we can consider the following 3-coskeletal simplicial set: there is exactly one vertex $\bullet$; there is an edge $\bullet\stackrel{g}{\to}\bullet$ for each element in $G$; there is a 2-simplex with edges $g_1,g_2,g_3$ and face $a\in A$ for any such elements such that $g_1g_2g_3=a$; there is a 3-simplex with faces $a_1,a_2,a_3,a_4$ (and elements from $G$ on the edges) whenever $\sum_i(-1)^i a_i=0$.
let me call $\mathbf{B}(G//A)$ this simplicial set. it should be a Kan complex (which is immediate, if I did not miss some point here) so it is the nerve of (the deloping of) a 2-group $G//A$. moreover there is a natural morphism $\mathbf{B}(G//A)\to \mathbf{B}H$ (a weak equivalence, I guess) which fits into a commutative diagram $\begin{matrix} \mathbf{B}(G//A)&\hookrightarrow& \mathbf{cosk}_1\mathbf{B}G\\ \downarrow &&\downarrow\\ \mathbf{B}H &\to &\mathbf{cosk}_1\mathbf{B}H \end{matrix}$
My thought is the following:
The inclusion of $A$ into $G$ is a crossed module (therefore gives a 2-group). The simplicial set you give is the clasifying space of that crossed module (bar one or two details). The other facts follow in general as this 2-group has trivial second homotopy group (= kernel).
There are numerous ways of looking at these things, e.g. classifying spaces of general crossed modules are discussed in the Menagerie, but that is not the only terminology one can use for these things.
Right, so that 2-group I’d notationally identify with the crossed module $(A \to G)$ as Tim says. This is a groupoid with $G$ as its objects and morphisms labeled by $A$. It gives a one-object 2-groupoid $\mathbf{B}(A \to G)$, yes.
The simplicial set you indicate is indeed the Duskin nerve $N \mathbf{B}(A \to G)$.
And, yes, the obvious 2-functor
$\mathbf{B}(A \to G) \to \mathbf{B}H$is an equivalence of 2-groupoids. You can check this for instance noticing that this is a k-surjective functor for all $k$, or equivalently of course by noticing that
$N\mathbf{B}(A \to G) \to N \mathbf{B}H$is an acyclic Kan fibration.
This weak equivalence, by the way, is the tool to extend the short exact sequence to the corresponding fiber sequence. In the $\infty$-category of $\infty$-groupoids this goes
$A \to G \to H \to \mathbf{B}A \to \mathbf{B}G \to \mathbf{B}H \to \mathbf{B}^2 A \,.$The last step is modeled in terms of strict functors by the 2-anafunctor
$\array{ \mathbf{B}(A \to G) &\to & \mathbf{B}(A \to 1) = \mathbf{B}^2 A \\ {}^{\mathrlap{\simeq}}\downarrow \\ \mathbf{B}H }$I have added this discussion to group extension – Central extensions
I have added this discussion to group extension – Central extensions
just looked at it: isn’t $\mathbf{B}(A\to G)$ 3-coskeletal? (and not 2-coskeletal, I mean)
Yes, thanks. I have fixed it now.
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