Explicit Factorization of a Categorical Center

Abstract

Given a braided fusion category C, it is well known that the natural map F: C ⊠ Cbop → Z(C) from the square of C to the (Drinfeld) categorical center Z(C) is an equivalence if and only if C is modular. This provides a non-constructive structure theorem for Z(C) for the modular case. However, it is not clear how to construct the inverse. In this work, we provide an explicit construction using insights from a specific quantum field theory. In particular, we construct an adjoint functor for F that is its inverse precisely when C is modular. The witnessing natural transformations are also constructed as values at certain cobordism of a specific 4-dimensional extended topological quantum field theory, the Crane-Yetter model. Such construction provides a (partial) factorization of the structure of Z(C) even when C is not modular. It is useful for understanding the extended structure of the Crane-Yetter model (future work). MSC: 18M20, 57K30, 57K40.