2023/07/12 by Nils Carqueville, Lukas Müller, Carqueville, Nils +1 · 2 citations
Mathematics · Medicine · #Advanced Topics in Algebra #Category Theory (math.CT) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Intracranial Aneurysms: Treatment and Complications #Mathematical Physics (math-ph) #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2307.06485
openalex publication_date 2023/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
We develop a general theory of 3-dimensional ``orbifold completion'', to describe (generalised) orbifolds of topological quantum field theories as well as all their defects. Given a semistrict 3-category T with adjoints for all 1- and 2-morphisms (more precisely, a Gray category with duals), we construct the 3-category T_\textrmorb as a Morita category of certain E1-algebras in T which encode triangulation invariance. We prove that in T_\textrmorb again all 1- and 2-morphisms have adjoints, that it contains T as a full subcategory, and we argue, but do not prove, that it satisfies a universal property which implies (T_\textrmorb)_\textrmorb ≅ T_\textrmorb. This is a categorification of the work in [CR]. Orbifold completion by design allows us to lift the orbifold construction from closed TQFT to the much richer world of defect TQFTs. We illustrate this by constructing a universal 3-dimensional state sum model with all defects from first principles, and we explain how recent work on defects between Witt equivalent Reshetikhin--Turaev theories naturally appears as a special case of orbifold completion.