2004/11/30 by Hendryk Pfeiffer · 2 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #math-ph #math.MP #math.QA #msc:16W30 #msc:18C10 #msc:18D05 #msc:18D10 #msc:20C15 #msc:22C05 #msc:46L05
paper · pdf · doi:10.1016/j.aim.2006.09.014
published as Adv. Math. 212 No. 1 (2007) 62-108 · 45 pages, LaTex2e with XYPic diagrams; v2: extended version; v3: typos corrected
arxiv created 2006/10/21 · openalex publication_date 2006/11/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.