Center (category theory)

Let ${\mathcal {C}}=({\mathcal {C}},\otimes ,I)$ be a (strict) monoidal category. The center of ${\mathcal {C}}$, also called the Drinfeld center of ${\mathcal {C}}$^[1] and 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

1. Majid 1991.

• Joyal, André; Street, Ross (1991), "Tortile Yang-Baxter operators in tensor categories", Journal of Pure and Applied Algebra, 71 (1): 43–51, doi:10.1016/0022-4049(91)90039-5, MR 1107651.
• Majid, Shahn (1991). "Representations, duals and quantum doubles of monoidal categories". Proceedings of the Winter School on Geometry and Physics (Srní, 1990). Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento (26). pp. 197–206. MR 1151906.
• Drinfeld center in nLab