Weyl integral

Let ${\mathcal {C}}=({\mathcal {C}},\otimes ,I)$ be a (strict) monoidal category. The centre of ${\mathcal {C}}$, denoted ${\mathcal {Z(C)}}$, is the category whose objects are pairs (A,u) consisting of an object A of ${\mathcal {C}}$ and a natural isomorphism $u_{X}:A\otimes X\rightarrow X\otimes A$ 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 ${\mathcal {Z(C)}}$ consists of an arrow $f:A\rightarrow B$ in ${\mathcal {C}}$ such that

$v_{X}(f\otimes 1_{X})=(1_{X}\otimes f)u_{X}$ .

The category ${\mathcal {Z(C)}}$ 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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