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.
such that This definition of the center appears in Joyal & Street (1991).
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.
becomes a braided monoidal category with the tensor product on objects defined as where
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
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
[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
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 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.
The center of a monoid and the Drinfeld center of a monoidal category are both instances of the following more general concept.
Similarly, if C is the category of abelian groups, monoid objects are rings, and the above recovers the center of a ring.
The categorical trace of a monoidal category (or monoidal ∞-category) is defined as The concept is being widely applied, for example in Zhu (2018).