2-Groups, trialgebras and their Hopf categories of representations

A strict 2-group is a 2-category with one object in which all morphisms and all 2-morphisms have inverses. 2-Groups have been studied in the context of homotopy theory, higher gauge theory and Topological Quantum Field Theory (TQFT). In the present article, we develop the notions of trialgebra and cotrialgebra, generalizations of Hopf algebras with two multiplications and one comultiplication or vice versa, and the notion of Hopf categories, generalizations of monoidal categories with an additional functorial comultiplication. We show that each strict 2-group has a ‘group algebra’ which is a cocommutative trialgebra, and that each strict finite 2-group has a ‘function algebra’ which is a commutative cotrialgebra. Each such commutative cotrialgebra gives rise to a symmetric Hopf category of corepresentations. In the semisimple case, this Hopf category is a 2-vector space according to Kapranov and Voevodsky. We also show that strict compact topological 2-groups are characterized by their C*-cotrialgebras of ‘complex-valued functions’, generalizing the Gel'fand representation, and that commutative cotrialgebras are characterized by their symmetric Hopf categories of corepresentations, generalizing Tannaka–Kreıˇn reconstruction. Technically, all these results are obtained using ideas from functorial semantics, by studying models of the essentially algebraic theory of categories in various base categories of familiar algebraic structures and the functors that describe the relationships between them.