Center (category theory) explained

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

l{C}=(l{C}, ⊗,I)

, denoted

l{Z(C)}

, is the category whose objects are pairs (A,u) consisting of an object A of

l{C}

and an isomorphism

uX:AXXA

which is natural in

X

satisfying

uX=(1uY)(uX1)

and

uI=1A

(this is actually a consequence of the first axiom).

An arrow from (A,u) to (B,v) in

l{Z(C)}

consists of an arrow

f:AB

in

l{C}

such that

vX(f1X)=(1Xf)uX

.This definition of the center appears in . Equivalently, the center may be defined as

lZ(lC)=

End
lClCop

(lC),

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

l{Z(C)}

becomes a braided monoidal category with the tensor product on objects defined as

(A,u)(B,v)=(AB,w)

where

wX=(uX1)(1vX)

, 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

ModR

of R-modules, for a commutative ring R, is

ModR

again. The center of a monoidal ∞-category C can be defined, analogously to the above, as

Z(lC):=

End
lClCop

(lC)

.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

E2

-monoidal category. More generally, the center of a

Ek

-monoidal category is an algebra object in

Ek

-monoidal categories and therefore, by Dunn additivity, an

Ek+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

oplusgVg

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.

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
AAop

(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

CCop

C.

The concept is being widely applied, for example in .

References