Orbifold completion of 3-categories

TL;DR

This paper develops a comprehensive theory of orbifold completion for 3-categories, enabling the description of orbifolds and defects in 3D topological quantum field theories, with applications to state sum models and defect theories.

Contribution

It constructs the orbifold completion as a Morita category of $E_1$-algebras in Gray categories with duals, extending orbifold concepts to defect TQFTs and providing a universal 3D state sum model.

Findings

Orbifold completion forms a 3-category with adjoints for all morphisms.

Contains the original 3-category as a full subcategory.

Connects to recent work on Witt equivalent Reshetikhin--Turaev theories.

Abstract

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 $\mathcal{T}$ with adjoints for all 1- and 2-morphisms (more precisely, a Gray category with duals), we construct the 3-category $\mathcal{T}_{\textrm{orb}}$ as a Morita category of certain $E_1$-algebras in $\mathcal{T}$ which encode triangulation invariance. We prove that in $\mathcal{T}_{\textrm{orb}}$ again all 1- and 2-morphisms have adjoints, that it contains $\mathcal{T}$ as a full subcategory, and we argue, but do not prove, that it satisfies a universal property which implies $(\mathcal{T}_{\textrm{orb}})_{\textrm{orb}} \cong \mathcal{T}_{\textrm{orb}}$. 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.