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 formula_1, denoted formula_2, is the category whose objects are pairs "(A,u)" consisting of an object "A" of formula_3 and an isomorphism formula_4 which is natural in formula_5 satisfying
and 
An arrow from "(A,u)" to "(B,v)" in formula_2 consists of an arrow formula_9 in formula_3 such that
This definition of the center appears in . Equivalently, the center may be defined as
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 formula_2 becomes a braided monoidal category with the tensor product on objects defined as
where formula_15, 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 formula_16 of "R"-modules, for a commutative ring "R", is formula_16 again. The center of a monoidal ∞-category "C" can be defined, analogously to the above, as
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 formula_19 (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 formula_20-monoidal category. More generally, the center of a formula_21-monoidal category is an algebra object in formula_21-monoidal categories and therefore, by Dunn additivity, an formula_23-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
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
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 
The concept is being widely applied, for example in .