2-Groups, trialgebras and their Hopf categories of representations

Abstract

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--Krein 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.

45 pages, LaTex2e with XYPic diagrams; v2: extended version; v3: typos corrected

References in corpus (8)

• Higher-Dimensional Algebra V: 2-Groups
• Higher Gauge Theory: 2-Connections on 2-Bundles
• Higher Yang-Mills Theory
• Higher gauge theory and a non-Abelian generalization of 2-form electrodynamics
• Categorical representations of categorical groups
• Group Objects and Internal Categories
• Representation theory of 2-groups on finite dimensional 2-vector spaces
• A More Sensitive Lorentzian State Sum

Cited by in corpus (7)

• Topological Higher Gauge Theory - from BF to BFCG theory
• A Prehistory of n-Categorical Physics
• Quantization via Linear homotopy types
• Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories
• 2-Drinfel'd double symmetry of the 4d Kitaev model
• A Categorification of Hall Algebras
• Fiat categorification of the symmetric inverse semigroup IS_n and the semigroup F^*_n