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 ${\displaystyle {\mathcal {C}}=({\mathcal {C}},\otimes ,I)}$, denoted ${\displaystyle {\mathcal {Z(C)}}}$, is the category whose objects are pairs ${\displaystyle (A,u)}$ consisting of an object ${\displaystyle A}$ of ${\displaystyle {\mathcal {C}}}$ and an isomorphism ${\displaystyle u_{X}:A\otimes X\rightarrow X\otimes A}$ which is natural in ${\displaystyle X}$ 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).^[1]

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

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

This definition of the center appears in Joyal & Street (1991). Equivalently, the center may be defined as

${\displaystyle {\mathcal {Z}}({\mathcal {C}})=\mathrm {End} _{{\mathcal {C}}\otimes {\mathcal {C}}^{op}}({\mathcal {C}}),}$

i.e., the endofunctors of ${\displaystyle {\mathcal {C}}}$ which are compatible with the left and right action of ${\displaystyle {\mathcal {C}}}$ on itself given by the tensor product.

Braiding

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.

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 ${\displaystyle \mathrm {Mod} _{R}}$ of ${\displaystyle R}$-modules, for a commutative ring ${\displaystyle R}$, is ${\displaystyle \mathrm {Mod} _{R}}$ again. The center of a monoidal ∞-category ${\displaystyle {\mathcal {C}}}$ can be defined, analogously to the above, as

${\displaystyle Z({\mathcal {C}}):=\mathrm {End} _{{\mathcal {C}}\otimes {\mathcal {C}}^{op}}({\mathcal {C}})}$.

Now, in contrast to the above, the center of the derived category of ${\displaystyle R}$-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 ${\displaystyle R}$ (as in the abelian situation above), but includes higher terms such as ${\displaystyle \mathrm {Hom} (R,R)}$ (derived Hom).^[2]

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 ${\displaystyle E_{2}}$-monoidal category. More generally, the center of a ${\displaystyle E_{k}}$-monoidal category is an algebra object in ${\displaystyle E_{k}}$-monoidal categories and therefore, by Dunn additivity, an ${\displaystyle E_{k+1}}$-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

${\displaystyle \bigoplus _{g\in G}V_{g}}$

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 ${\displaystyle {\mathcal {C}}}$ and a monoid object ${\displaystyle A}$ in ${\displaystyle {\mathcal {C}}}$, the center of ${\displaystyle A}$ is defined as

${\displaystyle Z(A)=\mathrm {End} _{A\otimes A^{\mathrm {op} }}(A).}$

For ${\displaystyle {\mathcal {C}}}$ being the category of sets (with the usual cartesian product), a monoid object is simply a monoid, and ${\displaystyle Z(A)}$ is the center of the monoid. Similarly, if ${\displaystyle {\mathcal {C}}}$ is the category of abelian groups, monoid objects are rings, and the above recovers the center of a ring. Finally, if ${\displaystyle {\mathcal {C}}}$ is the category of categories, with the product as the monoidal operation, monoid objects in ${\displaystyle {\mathcal {C}}}$ 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

${\displaystyle Tr(C):=C\otimes _{C\otimes C^{op}}C.}$

The concept is being widely applied, for example in Zhu (2018).