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Let $A,B$ objects of an abelian category. Then we can define the abelian group $\mathrm{Ext}^1(A,B)$ as the set of isomorphism classes of extensions $0 \to B \to E \to A \to 0$, endowed with the Baer sum. Following the principle of categorification, a finer and hopefully better invariant is the category of extensions, where morphisms are commutative diagrams as usual. Actually it is a groupoid by the Five Lemma, and I believe that the trivial extension $1$ and the Baer sum $\otimes$ make it a symmetric monoidal category in which every object has an inverse. In other words, it should be an abelian 2-group $\mathsf{Ext}^1(A,B)$, whose decategorification is the usual abelian group $\mathrm{Ext}^1(A,B)$. For this one has to find various coherence isomorphisms and check various coherence diagrams.

Question. Has this been worked out in the literature? Has this 2-group of extensions already been studied somewhere? What about specific examples?

Note that this 2-group carries more information than the group, for example the automorphism group of the unit $1$ is $\mathrm{Hom}(A,B)$. Here is an example with abelian groups: Let $p$ be a prime number. Then $\mathsf{Ext}^1(\mathbb{Z}/p,\mathbb{Z}/p)$ has $p$ non-isomorphic objects, namely the trivial extension $1$ and the extensions $X_\gamma : 0 \to \mathbb{Z}/p \xrightarrow{p \gamma} \mathbb{Z}/p^2 \xrightarrow{\text{pr}} \mathbb{Z}/p \to 0$ for $\gamma \in (\mathbb{Z}/p)^*$. We have $\mathrm{Aut}(1)=\mathbb{Z}/p$ and $\mathrm{Aut}(X_{\gamma}) = \{\alpha \in \mathbb{Z}/p^2 : \alpha \equiv 1 \bmod p\} \cong \mathbb{Z}/p$. Besides, $X_{\gamma} \otimes X_{\delta} \cong X_{\gamma + \delta}$.

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This 2-group arises within the Grothendieck's treatment of Mumford's idea of biextensions in the study of the relationship between Neron models of dual abelian varieties over fraction fields of discrete valuation rings; see Exposes VII, VIII in SGA 7, where Grothendieck denotes the category as EXT($A$, $B$); e.g., see Prop. 7.4(a) in Exp. VIII. See Breen's book "Fonctions theta et theoreme du cube" for a systematic consideration of the BIEXT (rather than just "Biext$^1$") perspective. – user36938 Jul 15 '13 at 12:21
@user36938: Thank you, why not making this an answer? – Martin Brandenburg Jul 15 '13 at 19:33
It doesn't answer your question, as Grothendieck doesn't work out anything with "coherence isomorphisms" (he has bigger fish to fry, so to speak). I just made the comment to point out a serious place not internal to category theory where this notion has naturally arisen long ago in the course of doing something which is of independent interest (for the arithmetic of abelian varieties). – user36938 Jul 16 '13 at 1:25
up vote 5 down vote accepted

A version of this, but one categorical level higher (i.e. a 3-group of (central) extensions of 2-groups) appeared explicitly in this paper of mine:

Central Extensions of smooth 2-groups and a finite dimensional string 2-group Geometry & Topology 15 (2011) 609-676.

Of course this is not the first time something like the 2-group you are interested in has appeared in the literature, but I don't have a specific reference. I don't know if there will be a good one because there is not much new information from this 2-group (see below).

Here are some observations about this "$EXT(A,B)$":

  • The fastest way to see that this is an abelian 2-group satisfying all the coherence data is to view $EXT(A,-)$ as a functor from abelian groups to groupoids. As such it sends abelian group objects in Abelian groups to abelian group objects in groupoids. Then note that every abelian group B is an abelian group object. This implies that all the coherence data is there.
  • There is a two term chain complex of abelian groups whose homology groups are $ext^1(A,B)$ and $Hom(A,B)$. There is a well-known process which turns a chain complex of abelain groups into a category internal to abelian groups and hence into a strictly commutative abelian group object in groupoids.
  • There is a functor from this chain-complex 2-group into the 2-group of actual extensions which sends elements from the complex to explicit extensions. This induces a functor, which is actually an equivalence of groupoids (it is enough to check an isom on $\pi_0$ and $\pi_1$), hence it is an equivalence of abelian 2-groups. (To see it is a functor of abelian 2-groups we just need to observe that all these constructions are functorial in B).

Either of the above two descriptions can also be used to show that all the k-invariants of this 2-group vanish. This means that this 2-group splits (non-canonically) as: $$ EXT(A,B) \simeq ext^1(A,B) \oplus \mathcal{B} hom(A,B) $$ where $EXT$ means the 2-group of extensions, $\pi_0 EXT = ext^1$ is the group of isomorphisms classes of extensions (thought of as a trivial 2-group), and "$\mathcal{B}$" means "shift the given abelian group up".

This means there is essentially no new information in this 2-group beyond its homotopy groups.

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Oh you are right, we have a canonical isomorphism $\mathrm{Aut}(E) \cong \mathrm{Hom}(A,B)$ for every extension $E$. – Martin Brandenburg Jul 15 '13 at 19:34

The paper Extensions of symmetric cat-groups by D. Bourn and E.M. Vitale defines and studies a bicategory of extensions of 2-groups (called cat-groups in the paper). In section 13, it introduces a monoidal structure that categorifies the Baer sum of extensions.

Just like Chris's answer, this is one categorical level higher then you wanted, but maybe you can decategorify it once :-)

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