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 $\mathcal = (\mathcal,\otimes,I)$, denoted $\mathcal$, is the category whose objects are pairs ''(A,u)'' consisting of an object ''A'' of $\mathcal$ and an isomorphism $u_X:A \otimes X \rightarrow X \otimes A$ which is natural in $X$ satisfying
: $u_ = (1 \otimes u_Y)(u_X \otimes 1)$

and

: $u_I = 1_A$ (this is actually a consequence of the first axiom).

An arrow from ''(A,u)'' to ''(B,v)'' in $\mathcal$ consists of an arrow $f:A \rightarrow B$ in $\mathcal$ 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

:$\mathcal Z(\mathcal C) = \mathrm_(\mathcal C),$

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 $\mathcal$ 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 $\mathrm_R$ of ''R''-modules, for a commutative ring ''R'', is $\mathrm_R$ again. The center of a monoidal ∞-category ''C'' can be defined, analogously to the above, as
:$Z(\mathcal C) := \mathrm_(\mathcal C)$.
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 $E_$-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
:$\bigoplus_ V_g$
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''.

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

:$Z(A) = End_(A).$

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
:$Tr(C) := C \otimes_ C.$
The concept is being widely applied, for example in .

References

*
*
*.
*
*
*

External links

*

{{Category theory

Category theory
Monoidal categories