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. 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. 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
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, 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.

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.