We introduce the notion of n-dimensional topological quantum field theory (TQFT) with defects as a symmetric monoidal functor on decorated stratified bordisms of dimension n. The familiar closed or open-closed TQFTs are special cases of defect TQFTs, and for n = 2 and n = 3 our general definition recovers what had previously been studied in the literature.Our main construction is that of "generalised orbifolds" for any n-dimensional defect TQFT: Given a defect TQFT Z, one obtains a new TQFT Z A by decorating the Poincaré duals of triangulated bordisms with certain algebraic data A and then evaluating with Z. The orbifold datum A is constrained by demanding invariance under n-dimensional Pachner moves. This procedure generalises both state sum models and gauging of finite symmetry groups, for any n. After developing the general theory, we focus on the case n = 3.

We obtain Morita invariant versions of Eilenberg-Watts type theorems, relating Deligne products of finite linear categories to categories of left exact as well as of right exact functors. This makes it possible to switch between different functor categories as well as Deligne products, which is often very convenient. For instance, we can show that applying the equivalence from left exact to right exact functors to the identity functor, regarded as a left exact functor, gives a Nakayama functor. The equivalences of categories we exhibit are compatible with the structure of module categories over finite tensor categories. This leads to a generalization of Radford's S 4 -theorem to bimodule categories. We also explain the relation of our construction to relative Serre functors on module categories that are constructed via inner Hom functors. G(m) ⊗ k m * = G(A) ,(1.3)

We consider traces on module categories over pivotal fusion categories which are compatible with the module structure. It is shown that such module traces characterise the Morita classes of special haploid symmetric Frobenius algebras. Moreover, they are unique up to a scale factor and they equip the dual category with a pivotal structure. This implies that for each pivotal structure on a fusion category over C there exists a conjugate pivotal structure defined by the canonical module trace. *

We initiate a systematic study of 3-dimensional 'defect' topological quantum field theories, that we introduce as symmetric monoidal functors on stratified and decorated bordisms. For every such functor we construct a tricategory with duals, which is the natural categorification of a pivotal bicategory. This captures the algebraic essence of defect TQFTs, and it gives precise meaning to the fusion of line and surface defects as well as their duality operations. As examples, we discuss how Reshetikhin-Turaev and Turaev-Viro theories embed into our framework, and how they can be extended to defect TQFTs.1

A modular tensor category C gives rise to a Reshetikhin-Turaev type topological quantum field theory which is defined on 3-dimensional bordisms with embedded C-coloured ribbon graphs. We extend this construction to include bordisms with surface defects which in turn can meet along line defects. The surface defects are labelled by ∆-separable symmetric Frobenius algebras and the line defects by "multi-modules" which are equivariant with respect to a cyclic group action. Our invariant cannot distinguish non-isotopic embeddings of 2-spheres, but we give an example where it distinguishes non-isotopic embeddings of 2-tori.