The Group Law of Picard Stacks via Matrices

An algebraic theory is a category whose objects are finite products of a distinguished object with itself. In this framework, abelian groups can be described as finite product-preserving functors from an appropriate algebraic theory to the category of sets. The purpose of this paper is to extend this categorical description of abelian groups to the setting of strictly commutative Picard stacks over a site S. More precisely, we show that such stacks can be characterized as morphisms of 2-stacks from a suitable algebraic 2-stack theory to the 2-stack of stacks over S. Our results provide a new formulation of the group law for strictly commutative Picard stacks, which may help clarify the notion of a group structure in higher stacks. We expect that this perspective will contribute further to the study of torsors, extensions, and biextensions in the higher categorical context.