With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
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.
We first give the general-abstract definition
of Drinfeld centers. Then we spell out what this means in components in
For a monoidal category, write for its delooping, the pointed 2-category with a single object such that .
The Drinfeld center of is the monoidal category of endo-pseudonatural transformations of the identity-2-functor on :
Unwinding the definitions, we find that an object of , , has for components pseudonaturality squares
for each . As shown, these consist of a choice of an object together with a natural isomorphism
in .
The transfor-property of says that
And so forth. Writing this out in terms of yields the following component characterization of Drinfeld centers, def. .
Let be a monoidal category. Its Drinfeld center is a monoidal category whose
objects are pairs of an object and a natural isomorphism (braiding morphism)
such that for all we have
morphisms are given by
the tensor product is given by
The Drinfeld center is naturally a braided monoidal category.
If is a fusion category over an algebraically closed field of characteristic zero, then the Drinfeld center 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.
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 algebra | category of modules |
---|---|
-algebra | -2-module |
sesquialgebra | 2-ring = monoidal presentable category with colimit-preserving tensor product |
bialgebra | strict 2-ring: monoidal category with fiber functor |
Hopf algebra | rigid monoidal category with fiber functor |
hopfish algebra (correct version) | rigid monoidal category (without fiber functor) |
weak Hopf algebra | fusion category with generalized fiber functor |
quasitriangular bialgebra | braided monoidal category with fiber functor |
triangular bialgebra | symmetric monoidal category with fiber functor |
quasitriangular Hopf algebra (quantum group) | rigid braided monoidal category with fiber functor |
triangular Hopf algebra | rigid symmetric monoidal category with fiber functor |
supercommutative Hopf algebra (supergroup) | rigid symmetric monoidal category with fiber functor and Schur smallness |
form Drinfeld double | form Drinfeld center |
trialgebra | Hopf monoidal category |
2-Tannaka duality for module categories over monoidal categories
monoidal category | 2-category of module categories |
---|---|
-2-algebra | -3-module |
Hopf monoidal category | monoidal 2-category (with some duality and strictness structure) |
3-Tannaka duality for module 2-categories over monoidal 2-categories
monoidal 2-category | 3-category of module 2-categories |
---|---|
-3-algebra | -4-module |
For a group, let denote the monoidal category of -graded vector spaces. Then the objects in consist of pairs for a finite conjugacy class of , along with a finite finite irreducible representation of the centralizer of . This is Example 8.5.4. in EGNO 2010.
Original articles:
Michael Mueger, From Subfactors to Categories and Topology II. The quantum double of tensor categories and subfactors, J. Pure Appl. Alg. 180, 159-219 (2003) (arXiv:math/0111205)
Pavel Etingof, Dmitri Nikshych, Victor Ostrik, On fusion categories, Annals of Mathematics Second Series, Vol. 162, No. 2 (Sep., 2005), pp. 581-642 (arXiv:math/0203060, jstor:20159926)
Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, On braided fusion categories I, Selecta Mathematica. New Series 16 (2010), no. 1, 1–119 (doi:10.1007/s00029-010-0017-z)
Review:
Textbook accounts:
Shahn Majid, Foundations of quantum group theory, Cambridge Univ. Press
Christian Kassel, Quantum groups, Graduate Texts in Mathematics 155, Springer 1995 (doi:10.1007/978-1-4612-0783-2, webpage, errata pdf)
Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik. (2010). Tensor Categories (Vol. 205). American Mathematical Soc.
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:
Last revised on October 30, 2024 at 14:57:01. See the history of this page for a list of all contributions to it.