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

${\displaystyle u_{X\otimes Y}=(1\otimes u_{Y})(u_{X}\otimes 1)}$

and

${\displaystyle u_{I}=1_{A}}$ (this is actually a consequence of the first axiom).^[1]

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

${\displaystyle v_{X}(f\otimes 1_{X})=(1_{X}\otimes f)u_{X}}$.

This definition of the center appears in Joyal & Street (1991). Equivalently, the center may be defined as

${\displaystyle {\mathcal {Z}}({\mathcal {C}})=\mathrm {End} _{{\mathcal {C}}\otimes {\mathcal {C}}^{op}}({\mathcal {C}}),}$

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

${\displaystyle (A,u)\otimes (B,v)=(A\otimes B,w)}$

where ${\displaystyle w_{X}=(u_{X}\otimes 1)(1\otimes v_{X})}$, 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 ${\displaystyle \mathrm {Mod} _{R}}$ of R-modules, for a commutative ring R, is ${\displaystyle \mathrm {Mod} _{R}}$ again. The center of a monoidal ∞-category C can be defined, analogously to the above, as

${\displaystyle Z({\mathcal {C}}):=\mathrm {End} _{{\mathcal {C}}\otimes {\mathcal {C}}^{op}}({\mathcal {C}})}$.

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 ${\displaystyle Hom(R,R)}$ (derived Hom).^[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 ${\displaystyle E_{2}}$-monoidal category. More generally, the center of a ${\displaystyle E_{k}}$-monoidal category is an algebra object in ${\displaystyle E_{k}}$-monoidal categories and therefore, by Dunn additivity, an ${\displaystyle E_{k+1}}$-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

${\displaystyle \bigoplus _{g\in G}V_{g}}$

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

In the same vein, Ben-Zvi, Francis & 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.

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

${\displaystyle Z(A)=End_{A\otimes A^{op}}(A).}$

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

${\displaystyle Tr(C):=C\otimes _{C\otimes C^{op}}C.}$

The concept is being widely applied, for example in Zhu (2018).

References

1. Majid 1991.
2. Ben-Zvi, Francis & Nadler (2010, Remark 1.5)

• 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, arXiv:0805.0157, doi:10.1090/S0894-0347-10-00669-7, MR 2669705, S2CID 2202294
• 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, arXiv:math/0511476, ISBN 978-0-8218-3713-9, Zbl 1142.18004
• Joyal, André; Street, Ross (1991), "Tortile Yang-Baxter operators in tensor categories", Journal of Pure and Applied Algebra, 71 (1): 43–51, doi:10.1016/0022-4049(91)90039-5, MR 1107651.
• Lurie, Jacob (2017), Higher Algebra
• Majid, Shahn (1991). "Representations, duals and quantum doubles of monoidal categories". Proceedings of the Winter School on Geometry and Physics (Srní, 1990). Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento. No. 26. pp. 197–206. hdl:10338.dmlcz/701494. MR 1151906.
• Zhu, Xinwen (2018). "Geometric Satake, categorical traces, and arithmetic of Shimura varieties". Current Developments in Mathematics. 2016: 145–206. arXiv:1810.07375. doi:10.4310/CDM.2016.v2016.n1.a4. ISBN 9781571463586. MR 3837875. OCLC 1038481072. S2CID 119589446.

External links

• Drinfeld center at the nLab