# Center (category theory)

In category theory, a branch of mathematics, the center (or Drinfeld center, after Soviet-American mathematician Vladimir Drinfeld) is a variant of the notion of the center of a monoid, group, or ring to a category.

## Definition

The center of a monoidal category ${\mathcal {C}}=({\mathcal {C}},\otimes ,I)$, denoted ${\mathcal {Z(C)}}$, is the category whose objects are pairs (A,u) consisting of an object A of ${\mathcal {C}}$ and an isomorphism $u_{X}:A\otimes X\rightarrow X\otimes A$ which is natural in $X$ 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
    $v_{X}(f\otimes 1_{X})=(1_{X}\otimes f)u_{X}$.
This definition of the center appears in Joyal & Street (1991). Equivalently, the center may be defined as
    ${\mathcal {Z}}({\mathcal {C}})=\mathrm {End} _{{\mathcal {C}}\otimes {\mathcal {C}}^{op}}({\mathcal {C}}),$
i.e., the endofunctors of C which are compatible with the left and right action of C on itself given by the tensor product.

### Braiding

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.

### Higher categorical version

The categorical center is particularly useful in the context of higher categories. This is illustrated by the following example: the center of the (abelian) category $\mathrm {Mod} _{R}$ of R-modules, for a commutative ring R, is $\mathrm {Mod} _{R}$ again. The center of a monoidal ∞-category C can be defined, analogously to the above, as
    $Z({\mathcal {C}}):=\mathrm {End} _{{\mathcal {C}}\otimes {\mathcal {C}}^{op}}({\mathcal {C}})$.
Now, in contrast to the above, the center of the derived category of R-modules (regarded as an ∞-category) is given by the derived category of modules over the cochain complex encoding the Hochschild cohomology, a complex whose degree 0 term is R (as in the abelian situation above), but includes higher terms such as $Hom(R,R)$ (derived Hom).
The notion of a center in this generality is developed by Lurie (2017, §5.3.1). Extending the above-mentioned braiding on the center of an ordinary monoidal category, the center of a monoidal ∞-category becomes an $E_{2}$-monoidal category. More generally, the center of a $E_{k}$-monoidal category is an algebra object in $E_{k}$-monoidal categories and therefore, by Dunn additivity, an $E_{k+1}$-monoidal category.

## Examples

Hinich (2007) has shown that the Drinfeld center of the category of sheaves on an orbifold X is the category of sheaves on the inertia orbifold of X. For X being the classifying space of a finite group G, the inertia orbifold is the stack quotient G/G, where G acts on itself by conjugation. For this special case, Hinich's result specializes to the assertion that the center of the category of G-representations (with respect to some ground field k) is equivalent to the category consisting of G-graded k-vector spaces, i.e., objects of the form
    $\bigoplus _{g\in G}V_{g}$
for some k-vector spaces, together with G-equivariant morphisms, where G acts on itself by conjugation.
In the same vein, Ben-Zvi, Francis & Nadler (2010) have shown that Drinfeld center of the derived category of quasi-coherent sheaves on a perfect stack X is the derived category of sheaves on the loop stack of X.

## Related notions
### Centers of monoid objects

The center of a monoid and the Drinfeld center of a monoidal category are both instances of the following more general concept. Given a monoidal category C and a monoid object A in C, the center of A is defined as
    $Z(A)=End_{A\otimes A^{op}}(A).$
For C being the category of sets (with the usual cartesian product), a monoid object is simply a monoid, and Z(A) is the center of the monoid. Similarly, if C is the category of abelian groups, monoid objects are rings, and the above recovers the center of a ring. Finally, if C is the category of categories, with the product as the monoidal operation, monoid objects in C are monoidal categories, and the above recovers the Drinfeld center.

### Categorical trace

The categorical trace of a monoidal category (or monoidal ∞-category) is defined as
    $Tr(C):=C\otimes _{C\otimes C^{op}}C.$
The concept is being widely applied, for example in Zhu (2018).