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, $X_{\gamma} \otimes X_{\delta} \cong X_{\gamma + \delta}$.