2-groups, trialgebras and their Hopf categories of representations (Q881959)

Starting from the group theory, one can develop the notions of: cocommutative and commutative Hopf algebra, symmetric monoidal category and so on. A \textit{strict \(2\)-group} is an internal category in the category of groups. The authors systematically replaces the word ``group'' by ``strict \(2\)-group'' to develop the notions of trialgebra, cotrialgebra and Hopf categories, generalizations of Hopf algebras and so on. It is shown that strict compact topological \(2\)-groups are characterized by their \(C^\ast\)-cotrialgebras of ``complex-valued functions'', generalizing the Gelfand representation, and that commutative cotrialgebras are characterized by their symmetric Hopf categories corepresentations, generalizing Tannaka-Kreîn reconstruction.