Center (category theory)

Short description: Variant of the notion of the center of a monoid, group, or ring to a category

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 ${𝒞}=({𝒞},\otimes ,I)$, denoted ${𝒵}{(}{𝒞}{)}$, is the category whose objects are pairs (A,u) consisting of an object A of ${𝒞}$ and an isomorphism $u_X:A\otimes X\to X\otimes A$ 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 ${𝒵}{(}{𝒞}{)}$ consists of an arrow $f:A\to B$ in ${𝒞}$ such that

$v_X(f\otimes 1_X)=(1_X\otimes f)u_X$.

This definition of the center appears in (Joyal Street). Equivalently, the center may be defined as

${𝒵}({𝒞})={\mathrm{E}\mathrm{n}\mathrm{d}}_{{𝒞}\otimes {{𝒞}}^{op}}({𝒞}),$

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 ${𝒵}{(}{𝒞}{)}$ 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 ${\mathrm{M}\mathrm{o}\mathrm{d}}_R$ of R-modules, for a commutative ring R, is ${\mathrm{M}\mathrm{o}\mathrm{d}}_R$ again. The center of a monoidal ∞-category C can be defined, analogously to the above, as

$Z({𝒞}):={\mathrm{E}\mathrm{n}\mathrm{d}}_{{𝒞}\otimes {{𝒞}}^{op}}({𝒞})$.

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 (Lurie 2017). 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

(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

$\underset{g\in G}{⨁}{V}_g$

for some k-vector spaces, together with G-equivariant morphisms, where G acts on itself by conjugation.

In the same vein, (Ben-Zvi Francis) 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

$Z(A)=En{d}_{A\otimes A^{op}}(A).$

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 (Zhu 2018).

References

• Ben-Zvi, David; Francis, John; Nadler, David (2010), "Integral transforms and Drinfeld centers in derived algebraic geometry", Journal of the American Mathematical Society 23 (4): 909–966, doi:10.1090/S0894-0347-10-00669-7 
• Hinich, Vladimir (2007), "Drinfeld double for orbifolds", Israel mathematical conference proceedings. Quantum groups. Proceedings of a conference in memory of Joseph Donin, Haifa, Israel, July 5--12, 2004, AMS, pp. 251–265, ISBN 978-0-8218-3713-9 
• "Tortile Yang-Baxter operators in tensor categories", Journal of Pure and Applied Algebra 71 (1): 43–51, 1991, doi:10.1016/0022-4049(91)90039-5 .
• Lurie, Jacob (2017), Higher Algebra, http://www.math.harvard.edu/~lurie/ 
• "Representations, duals and quantum doubles of monoidal categories". 1991. 197–206. https://eudml.org/doc/220868. 
• Zhu, Xinwen (2018). "Geometric Satake, categorical traces, and arithmetic of Shimura varieties". Current Developments in Mathematics 2016: 145–206. doi:10.4310/CDM.2016.v2016.n1.a4. ISBN 9781571463586. OCLC 1038481072. 

External links

• Drinfeld center in nLab