# Morita invariance of Drinfeld centre

Given a monoidal category $M$ one can consider its Drinfeld centre $Z(M)$. Objects of the Drinfeld centre are pairs $(m, \alpha)$ where $m$ is an object and $\alpha$ is an isomorphism $\alpha: - \otimes m \to m \otimes -$ satisfying some "obvious" conditions.

A simple and important example of a monoidal category is the category of $G$-equivariant sheaves of $k$-vector spaces on $Y \times Y$ where $Y$ is a finite $G$-set. The monoidal structure is given by

$V\otimes W = p_{13*} (p_{12}^*V \otimes p_{23}^* W)$

where $p_{ij} : Y \times Y \times Y \to Y \times Y$ denotes the projection map (in a hopefully obvious notation).

For example, if $Y$ is a point then one recovers the (tensor) category of representations of $G$. If $Y = G$ then one recovers a monoidal category equivalent vectors spaces graded by $G$. If $G$ is the trivial group then one obtains the tensor category of matrices of vector spaces'' over $Y$.

Now there is a result, which I have heard (by Ostrik) called Muerger's Morita invariance of Drinfeld centre''. It should have the consequence that, with $G$ and $Y$ as above:

The Drinfeld centre $Z(Sh_G(Y \times Y))$ does not depend on $Y$ up to equivalence.

(I guess the baby example of $G$ the trivial group explains the term Morita invariance''.)

My question is:

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In your case, if $Y$ and $Y'$ are two $G$-sets, then $Sh_G(Y \times Y')$ is a bimodule from $Sh_G(Y \times Y)$ to $Sh_G(Y' \times Y')$ with the obvious left and right actions. This bimodule is invertible, with inverse $Sh_G(Y' \times Y)$. In other words, $$Sh_G(Y \times Y') \boxtimes_{Sh_G(Y' \times Y')} Sh_G(Y' \times Y) \cong Sh_G(Y \times Y)$$ as $Sh_G(Y \times Y)$-bimodules, and similarly with $Y$ and $Y'$ switched.