Keeley Hoek's 2019 honours thesis in mathematics, on "Drinfeld Centers for Bimodule Categories", supervised by

Scott Morrison, is available as

thesis.pdf.

### Outline

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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 $\mathcal{Z}(\mathcal{C})$.