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Drinfeld center in nLab

nLab Drinfeld center

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Contents

Context

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

Contents

Idea

The notion of center of a monoidal category or Drinfeld center is the categorification of the notion of center of a monoid(associative algebra, group, etc.) from monoids to monoidal categories.

Where the center of a monoid is just a sub-monoid with the property that it commutes with everything else, under categorification this becomes a structure, since we have to specify how the objects in the Drinfeld center commute (braid) with everything else.

Definition

We first give the general-abstract definition

of Drinfeld centers. Then we spell out what this means in components in

Abstractly

Definition

For (π’ž,βŠ—)(\mathcal{C}, \otimes) a monoidal category, write B βŠ—π’ž\mathbf{B}_\otimes \mathcal{C} for its delooping, the pointed 2-category with a single object ** such that Hom B βŠ—π’ž(*,*)β‰ƒπ’žHom_{\mathbf{B}_\otimes \mathcal{C}}(*, *) \simeq \mathcal{C}.

The Drinfeld center Z(π’ž,βŠ—)Z(\mathcal{C}, \otimes) of (π’ž,βŠ—)(\mathcal{C}, \otimes) is the monoidal category of endo-pseudonatural transformations of the identity-2-functor on B βŠ—π’ž\mathbf{B}_\otimes \mathcal{C}:

Z(π’ž,βŠ—)≔End B βŠ—π’ž(id B βŠ—π’ž). Z(\mathcal{C}, \otimes) \coloneqq End_{\mathbf{B}_\otimes \mathcal{C}}(id_{\mathbf{B}_\otimes \mathcal{C}}) \,.
Remark

Unwinding the definitions, we find that an object of Z(π’ž,βŠ—)Z(\mathcal{C}, \otimes), Ξ¦:id B βŠ—π’žβ†’id B βŠ—π’ž\Phi \colon id_{\mathbf{B}_\otimes \mathcal{C}} \to id_{\mathbf{B}_\otimes \mathcal{C}}, has for components pseudonaturality squares

* β†’X≔Φ(*) * Y↓ ⇙ Ξ¦ Y ↓ Y * β†’X=Ξ¦(*) * \array{ \ast &\stackrel{X \coloneqq \Phi(\ast)}{\to}& \ast \\ {}^{\mathllap{Y}}\downarrow &\swArrow_{\Phi_Y}& \downarrow^{\mathrlap{Y}} \\ \ast &\underset{X = \Phi(\ast)}{\to}& \ast }

for each Y∈Obj(π’ž)Y \in Obj(\mathcal{C}). As shown, these consist of a choice of an object Xβˆˆπ’žX \in \mathcal{C} together with a natural isomorphism

Ξ¦ (βˆ’):XβŠ—(βˆ’)β†’(βˆ’)βŠ—X \Phi_{(-)} \colon X \otimes (-) \to (-) \otimes X

in π’ž\mathcal{C}.

The transfor-property of Ξ¦\Phi says that

* β†’X≔Φ(*) * Y↓ ⇙ Ξ¦ Y ↓ Y * β†’X=Ξ¦(*) * Z↓ ⇙ Ξ¦ Z ↓ Z * β†’X=Ξ¦(*) *≃* β†’X≔Φ(*) * YβŠ—Z↓ ⇙ Ξ¦ YβŠ—Z ↓ YβŠ—Z * β†’X=Ξ¦(*) *. \array{ \ast &\stackrel{X \coloneqq \Phi(\ast)}{\to}& \ast \\ {}^{\mathllap{Y}}\downarrow &\swArrow_{\Phi_Y}& \downarrow^{\mathrlap{Y}} \\ \ast &\underset{X = \Phi(\ast)}{\to}& \ast \\ {}^{\mathllap{Z}}\downarrow &\swArrow_{\Phi_Z}& \downarrow^{\mathrlap{Z}} \\ \ast &\underset{X = \Phi(\ast)}{\to}& \ast } \;\;\;\; \simeq \;\;\;\; \;\;\;\; \array{ \ast &\stackrel{X \coloneqq \Phi(\ast)}{\to}& \ast \\ {}^{\mathllap{Y \otimes Z}}\downarrow &\swArrow_{\Phi_{Y \otimes Z}}& \downarrow^{\mathrlap{Y \otimes Z}} \\ \ast &\underset{X = \Phi(\ast)}{\to}& \ast } \,.

And so forth. Writing this out in terms of (π’ž,βŠ—)(\mathcal{C}, \otimes) yields the following component characterization of Drinfeld centers, def. .

In components

Definition

Let (π’ž,βŠ—)(\mathcal{C}, \otimes) be a monoidal category. Its Drinfeld center is a monoidal category Z(π’ž)Z(\mathcal{C}) whose

  • objects are pairs (X,Ξ¦)(X, \Phi) of an object Xβˆˆπ’žX \in \mathcal{C} and a natural isomorphism (braiding morphism)

    Ξ¦:XβŠ—(βˆ’)β†’(βˆ’)βŠ—X \Phi \colon X \otimes (-) \to (-) \otimes X

    such that for all Yβˆˆπ’žY \in \mathcal{C} we have

    Ξ¦ YβŠ—Z=(idβŠ—Ξ¦ Z)∘(Ξ¦ YβŠ—id) \Phi_{Y \otimes Z} = (id \otimes \Phi_Z) \circ (\Phi_Y \otimes id)

  • morphisms are given by

    Hom((X,Ξ¦),(Y,Ξ¨))={f∈Hom π’ž(X,Y)|(idβŠ—f)∘Φ Z=Ξ¨ Z∘(fβŠ—id),βˆ€Zβˆˆπ’ž}. Hom((X, \Phi), (Y,\Psi)) = \left\{ f \in Hom_{\mathcal{C}}(X,Y) \;|\; (id \otimes f) \circ \Phi_Z = \Psi_Z \circ (f \otimes id), \; \forall Z \in \mathcal{C} \right\} \,.

  • the tensor product is given by

    (X,Ξ¦)βŠ—(Y,Ξ¨)=(XβŠ—Y,(Ξ¦βŠ—id)∘(idβŠ—Ξ¨)). (X, \Phi) \otimes (Y, \Psi) = (X \otimes Y, (\Phi \otimes id) \circ (id \otimes \Psi)) \,.

Properties

Extra structure on the Drinfeld center

Proposition

The Drinfeld center Z(π’ž)Z(\mathcal{C}) is naturally a braided monoidal category.

Proposition

If π’ž\mathcal{C} is a fusion category over an algebraically closed field of characteristic zero, then the Drinfeld center Z(π’ž)Z(\mathcal{C}) is also naturally a fusion category.

(Etingof, Nikshych & Ostrik 2005, Thm. 2.15, review in Davydov, Mueger, Nikshych & Ostrik 2003, Sec. 2.3)

See also Drinfeld, Gelaki, Nikshych & Ostrik 2010, Cor. 3.9, Mueger 2003.

Relation to Drinfeld double under Tannaka duality

Under Tannaka duality, forming the Drinfeld center of a category of modules of some Hopf algebra corresponds to forming the category of modules over the corresponding Drinfeld double algebra. See there for more.

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebracategory of modules
AAMod AMod_A
RR-algebraMod RMod_R-2-module
sesquialgebra2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebrastrict 2-ring: monoidal category with fiber functor
Hopf algebrarigid monoidal category with fiber functor
hopfish algebra (correct version)rigid monoidal category (without fiber functor)
weak Hopf algebrafusion category with generalized fiber functor
quasitriangular bialgebrabraided monoidal category with fiber functor
triangular bialgebrasymmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)rigid braided monoidal category with fiber functor
triangular Hopf algebrarigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld doubleform Drinfeld center
trialgebraHopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category2-category of module categories
AAMod AMod_A
RR-2-algebraMod RMod_R-3-module
Hopf monoidal categorymonoidal 2-category (with some duality and strictness structure)

3-Tannaka duality for module 2-categories over monoidal 2-categories

monoidal 2-category3-category of module 2-categories
AAMod AMod_A
RR-3-algebraMod RMod_R-4-module

Examples

Vec GVec_G

For GG a group, let Vec GVec_G denote the monoidal category of GG-graded vector spaces. Then the objects in 𝒡(Vec G)\mathcal{Z}(Vec_G ) consist of pairs (C,V)(C,V) for CC a finite conjugacy class of GG, along with VV a finite finite irreducible representation of the centralizer of g∈Cg\in C. This is Example 8.5.4. in EGNO 2010.

References

Original articles:

Review:

Textbook accounts:

A general discussion of centers of monoid objects in braided monoidal 2-categories (which reduces to the above for the 2-category Cat with its cartesian product) is in

An application to character sheaves is in

In relation to spectra of tensor triangulated categories:

Relation to Frobenius monoidal functors:

  • Johannes Flake, Robert Laugwitz, Sebastian Posur: Frobenius monoidal functors from ambiadjunctions and their lifts to Drinfeld centers [arXiv:2410.08702]

Last revised on October 30, 2024 at 14:57:01. See the history of this page for a list of all contributions to it.

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