Drinfeld Centers for Bimodule Categories Keeley Hoek

Keeley Hoek's 2019 honours thesis in mathematics, on "Drinfeld Centers for Bimodule Categories", supervised by Scott Morrison, is available as thesis.pdf.



We prove the folklore result that the Drinfeld center $\mathcal{Z}(\mathcal{C})$ of a pivotal category $\mathcal{C}$ is contravariantly equivalent (as a braided monoidal category) to the category of representations of the annular category of $\mathcal{C}$, when $\mathcal{C}$ is finitely semisimple. This has only been briefly sketched in the literature previously.

Given bimodule categories $\sub{\mathcal{D}}{\mathcal{M}}{\mathcal{C}}$ and $\sub{\mathcal{C}}{\mathcal{N}}{\mathcal{D}}$ under mild hypotheses there are two general constructions we can make; first, we can form their Deligne product $\sub{}{\mathcal{M}}{} \underset{\mathcal{C} \boxtimes \mathcal{D}^\mathrm{mop}}{\boxtimes} \sub{}{\mathcal{N}}{}$, which is a purely algebraic object. We can also consider representations of a "bimodule annular category", consisting of diagrams drawn in the annulus with equatorial boundaries labelled by $\sub{\mathcal{D}}{\mathcal{M}}{\mathcal{C}}$ and $\sub{\mathcal{C}}{\mathcal{N}}{\mathcal{D}}$. We prove that the Deligne product and representations of the bimodule annular category are contravariantly equivalent in the finitely semisimple case. As corollaries we deduce characterisations of the bimodule Drinfeld center $\mathcal{Z}(\sub{\mathcal{C}}{\mathcal{M}}{\mathcal{C}})$ and module functor categories $[\sub{\mathcal{C}}{\mathcal{N}}{\mathcal{D}} \to \sub{\mathcal{C}}{\mathcal{K}}{\mathcal{D}}]$, each as representations of a special case of the bimodule annular category. To prove these equivalences we introduce a notion of "balanced tensor products" of module categories, which we show in particular gives a new model for the Deligne product $\mathcal{M} \underset{\mathcal{C}}{\boxtimes} \mathcal{N}$ in the finitely semisimple case.

Finally, we use the balanced tensor product to define the notion of a bibalanced center $\mathcal{Z}^\mathrm{bibal}(\sub{\mathcal{C}}{\mathcal{M}}{\mathcal{D}})$ of an arbitrary bimodule category $\sub{\mathcal{C}}{\mathcal{M}}{\mathcal{D}}$, for which we establish a monoidal structure generalising the monoidal product in $\mathcal{Z}(\mathcal{C})$.

last modified: 2019-10-25