Center (category theory)

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Let ${\displaystyle {\mathcal {C}}=({\mathcal {C}},\otimes ,I)}$ be a (strict) monoidal category. The center of ${\displaystyle {\mathcal {C}}}$, also called the Drinfeld center of ${\displaystyle {\mathcal {C}}}$[1] and denoted ${\displaystyle {\mathcal {Z(C)}}}$, is the category whose objects are pairs (A,u) consisting of an object A of ${\displaystyle {\mathcal {C}}}$ and a natural isomorphism ${\displaystyle u_{X}:A\otimes X\rightarrow X\otimes A}$ satisfying

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

and

${\displaystyle u_{I}=1_{A}}$ (this is actually a consequence of the first axiom).

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

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

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

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

where ${\displaystyle w_{X}=(u_{X}\otimes 1)(1\otimes v_{X})}$, and the obvious braiding .