Weyl integral

Let $\mathcal{𝒞}=(\mathcal{𝒞},\otimes ,I)$ be a (strict) monoidal category. The centre of $\mathcal{𝒞}$, denoted $\mathcal{𝒵}\mathcal{(}\mathcal{𝒞}\mathcal{)}$, is the category whose objects are pairs (A,u) consisting of an object A of $\mathcal{𝒞}$ and a natural isomorphism $u_X:A\otimes X\to X\otimes A$ satisfying

$u_{X\otimes Y}=(1\otimes u_Y)(u_X\otimes 1)$

and

$u_I=1_A$ (this is actually a consequence of the first axiom).

An arrow from (A,u) to (B,v) in $\mathcal{𝒵}\mathcal{(}\mathcal{𝒞}\mathcal{)}$ consists of an arrow $f:A\to B$ in $\mathcal{𝒞}$ such that

$v_X(f\otimes 1_X)=(1_X\otimes f)u_X$ .

The category $\mathcal{𝒵}\mathcal{(}\mathcal{𝒞}\mathcal{)}$ becomes a braided monoidal category with the tensor product on objects defined as

$(A,u)\otimes (B,v)=(A\otimes B,w)$

where $w_X=(u_X\otimes 1)(1\otimes v_X)$, and the obvious braiding .

References

André Joyal and Ross Street. Tortile Yang-Baxter operators in tensor categories, J. Pure Appl. Algebra 71 (1991): 43–51.

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