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 [math]\displaystyle{ \mathcal{C} = (\mathcal{C},\otimes,I) }[/math], denoted [math]\displaystyle{ \mathcal{Z(C)} }[/math], is the category whose objects are pairs (A,u) consisting of an object A of [math]\displaystyle{ \mathcal{C} }[/math] and an isomorphism [math]\displaystyle{ u_X:A \otimes X \rightarrow X \otimes A }[/math] which is natural in [math]\displaystyle{ X }[/math] satisfying

[math]\displaystyle{ u_{X \otimes Y} = (1 \otimes u_Y)(u_X \otimes 1) }[/math]

and

[math]\displaystyle{ u_I = 1_A }[/math] (this is actually a consequence of the first axiom).

An arrow from (A,u) to (B,v) in [math]\displaystyle{ \mathcal{Z(C)} }[/math] consists of an arrow [math]\displaystyle{ f:A \rightarrow B }[/math] in [math]\displaystyle{ \mathcal{C} }[/math] such that

[math]\displaystyle{ v_X (f \otimes 1_X) = (1_X \otimes f) u_X }[/math].

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

[math]\displaystyle{ \mathcal Z(\mathcal C) = \mathrm{End}_{\mathcal C \otimes \mathcal C^{op}}(\mathcal C), }[/math]

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 [math]\displaystyle{ \mathcal{Z(C)} }[/math] becomes a braided monoidal category with the tensor product on objects defined as

[math]\displaystyle{ (A,u) \otimes (B,v) = (A \otimes B,w) }[/math]

where [math]\displaystyle{ w_X = (u_X \otimes 1)(1 \otimes v_X) }[/math], 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 [math]\displaystyle{ \mathrm{Mod}_R }[/math] of R-modules, for a commutative ring R, is [math]\displaystyle{ \mathrm{Mod}_R }[/math] again. The center of a monoidal ∞-category C can be defined, analogously to the above, as

[math]\displaystyle{ Z(\mathcal C) := \mathrm{End}_{\mathcal C \otimes \mathcal C^{op}}(\mathcal C) }[/math].

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 [math]\displaystyle{ Hom(R, R) }[/math] (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 [math]\displaystyle{ E_2 }[/math]-monoidal category. More generally, the center of a [math]\displaystyle{ E_k }[/math]-monoidal category is an algebra object in [math]\displaystyle{ E_k }[/math]-monoidal categories and therefore, by Dunn additivity, an [math]\displaystyle{ E_{k+1} }[/math]-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

[math]\displaystyle{ \bigoplus_{g \in G} V_g }[/math]

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

[math]\displaystyle{ Z(A) = End_{A \otimes A^{op}}(A). }[/math]

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

[math]\displaystyle{ Tr(C) := C \otimes_{C \otimes C^{op}} C. }[/math]

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