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.

The center of a monoidal category ${\displaystyle {𝒞}=({𝒞},\otimes ,I)}$, denoted ${\displaystyle {𝒵}{(}{𝒞}{)}}$, is the category whose objects are pairs (A,u) consisting of an object A of ${\displaystyle {𝒞}}$ and an isomorphism ${\displaystyle u_X:A\otimes X\to X\otimes A}$ 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 (A,u) to (B,v) in ${\displaystyle {𝒵}{(}{𝒞}{)}}$ consists of an arrow ${\displaystyle f:A\to B}$ in ${\displaystyle {𝒞}}$ such that

${\displaystyle v_X(f\otimes 1_X)=(1_X\otimes f)u_X}$.

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

${\displaystyle {𝒵}({𝒞})={\mathrm{E}\mathrm{n}\mathrm{d}}_{{𝒞}\otimes {{𝒞}}^{op}}({𝒞}),}$

i.e., the endofunctors of C which are compatible with the left and right action of C on itself given by the tensor product.

The category ${\displaystyle {𝒵}{(}{𝒞}{)}}$ 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.

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

${\displaystyle Z({𝒞}):={\mathrm{E}\mathrm{n}\mathrm{d}}_{{𝒞}\otimes {{𝒞}}^{op}}({𝒞})}$.

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 ${\displaystyle Hom(R,R)}$ (derived Hom).^[2]

The notion of a center in this generality is developed by (Lurie 2017). 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.

(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

${\displaystyle \underset{g\in G}{⨁}{V}_g}$

for some k-vector spaces, together with G-equivariant morphisms, where G acts on itself by conjugation.

In the same vein, (Ben-Zvi Francis) 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.

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

${\displaystyle Z(A)=En{d}_{A\otimes A^{op}}(A).}$

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.

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).

• Drinfeld center in nLab

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