On dualizability of braided tensor categories

Abstract

We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided multi-fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively two-, three- and four-dimensional framed local topological field theories. In particular, we produce a framed three-dimensional local topological field theory attached to the category of representations of a quantum group at any value of $q$ .

A categorical approach to dynamical quantum groups

“…It is clear that this isomorphism preserves coproduct, counit and unit and the only nontrivial check is that the product is preserved as well. The relations (18), (19) in 31 are clearly satisfied. For (20), the claim follows from Theorem 5.22.…”

“…Since is monoidal, the objects are also dualisable. But we have just shown that such objects are the generating compact projective objects, while by [18, Proposition 4.1], it is enough to check cp-rigidity on the generating compact projective objects.…”

“…If C is cp-rigid, the tensor product functor 𝑇 : C ⊗ C → C admits a colimit-preserving right adjoint 𝑇 R : C → C ⊗ C (see, e.g., [18,Section 5.3]). It has the following explicit formula.…”

“…Consider C ⊗ C as a C ⊗ C ⊗op -module category via the left action on the first factor and the right action on the second factor. By [18,Proposition 4.1], 𝑇 R is a functor of C ⊗ C ⊗op -module categories. This can be expressed in the following isomorphism.…”

The finiteness conjecture for skein modules

“…In that setting, the 4-dimensional Crane-Yetter-Kauffman TFT carries a boundary theory given by the 3dimensional Witten-Reshetikhin-Turaev TFT, a mathematical incarnation of Chern-Simons theory for the compact form of G. It seems natural to view Walker's TFT associated to the ribbon category Rep q (G) with q generic in the context of analytically continued Chern-Simons theory as discussed in [95]. (2) The work [18] constructs a 3-2-1-0 TFT for an arbitrary rigid braided tensor category. It is conjectured there that for semi-simple ribbon categories their construction coincides with Walker's 3-2 TFT.…”