Singular extensions and the second cohomology categorical group (Q2363410)

The aim of the article is to give a cohomological classification of the symmetric categorical group of singular extensions of a categorical group \(\mathbb G\) by a given symmetric \(\mathbb G\)-categorical group \(\mathbb A\). A \textit{categorical group} (also called gr-category or \(2\)-group) is a monoidal groupoid in which every element is invertible. It is called \textit{symmetric} if it is symmetric as a monoidal category. In order to get such classification, the author introduces a second cohomology categorical group \(\mathcal H^2(\mathbb G,\mathbb A)\), obtained from a cochain complex of symmetric categorical groups built through the nerve of \(\mathbb G\), a simplicial set. The author then proves the existence of a biequivalence of bigroupoids, giving the desired classification result. In addition, the author determines a Baer sum in the groupoid of singular extensions. Building up on previous work, the author applies these new results and proves categorical group versions of the classical Hochschild-Serre \(5\)-term exact group sequences involving the second cohomology categorical group introduced in the present paper. These sequences are associated to any essentially surjective homomorphism of categorical groups, and, in them, new categorical groups of ``non-functorial'' derivations appear.