Orbifold Completion of 3-Categories

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 T with adjoints for all 1- and 2-morphisms (more precisely, a Gray category with duals), we construct the 3-category Torb T orb as a Morita category of certain E₁ E 1 -algebras in T T which encode triangulation invariance. We prove that in Torb T orb again all 1- and 2-morphisms have adjoints, that it contains T T as a full subcategory, and we argue, but do not prove, that it satisfies a universal property which implies (Torb)orb Torb ( T orb ) orb ≅ T orb . This is a categorification of the work in Carquevill and Runkel (Quantum Topol 7(2):203–279, 2016). 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.