Weyl integral

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

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

.

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

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

References

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