Centre (category)

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 $u\_X:A\; otimes\; X\; ightarrow\; X\; otimes\; A$ 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 $f:A\; ightarrow\; B$ 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.