The center functor is fully faithful
Liang Kong, Hao Zheng
TL;DR
The paper resolves functoriality and faithfulness of the Drinfeld center in a categorical setting by proving that $\mathfrak{Z}$ extends to a symmetric monoidal functor from indecomposable multi-tensor categories to braided tensor categories, with module morphisms given by bimodules. It then proves that, when restricted to indecomposable multi-fusion categories, this center functor is fully faithful, and derives key consequences: Morita equivalence of fusion categories is detected by braided equivalence of their centers, and the group of invertible bimodules corresponds to braided auto-equivalences of centers. The methods fuse categorical constructions (tensor products of module categories, rigidity criteria, central functors) with physical insights from topological orders to give a rigorous boundary-bulk correspondence in 2+1D anomaly-free systems with gapped boundaries. Together, these results provide new, conceptually unified proofs of foundational statements in fusion category theory and deepen the connection between categorical centers and boundary phenomena in physics.
Abstract
We prove that the notion of Drinfeld center defines a functor from the category of indecomposable multi-tensor categories with morphisms given by bimodules to that of braided tensor categories with morphisms given by monoidal bimodules. Moreover, we apply some ideas from the physics of topological orders to prove that the center functor restricted to indecomposable multi-fusion categories (with additional conditions on the target category) is fully faithful. As byproducts, we provide new proofs to some important known results in fusion categories. In physics, this fully faithful functor gives the precise mathematical description of the boundary-bulk relation for 2+1D anomaly-free topological orders with gapped boundaries.