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](http://ncatlab.org/nlab/show/vertical+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](http://arxiv.org/abs/math/0307200) $\mathsf{Ext}^1(A,B)$, 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, $X\_{\gamma} \otimes X\_{\delta} \cong X\_{\gamma + \delta}$.



The 2-group of extensions
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](http://arxiv.org/abs/0911.2483) 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. 


