2025/01/14 by Hank Chen, Chen, Hank
Mathematics · #17B37 #18M15 (Priamry) #18N25 (Secondary) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2501.08041
openalex publication_date 2025/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a companion work on the combinatorial quantization of 4d 2-Chern-Simons theory, the author has constructed the Hopf category of quantum 2-gauge transformations C=\mathbbUq\mathfrakG acting on the discrete surface-holonomy configurations on a lattice. We prove in this article that the 2-Hilb-enriched 2-representation 2-category 2Rep( C) of finite semisimple ℂ-linear C-module categories is braided, planar-pivotal, and lax rigid, hence 2Rep( C) provides an example of a ribbon tensor 2-category. We explicitly construct the ribbon balancing functors, and exhibit their coherence conditions against the rigid dagger structures. This allows one to refine the various notions of framing in a 2-category with duals that have been previously studied in the literature. Following the 2-tangle hypothesis of Baez-Langford, framed invariants of 2-tangles can then be constructed from ribbon 2-functors into 2Rep( C), analogous to the definition of decorated ribbon graphs in the Reshetikhin-Turaev construction. We will also prove that, in the classical limit q→ 1, the 2-category 2Rep(\mathbbUq=1\mathfrakG) becomes strict pivotal in the sense of Douglas-Reutter.