nLab
$G$-crossed braided fusion category

Contents

Context

Monoidal categories

Fusion categories

monoidal categories

With symmetry

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

Contents

Intuition

A fusion category over a field kk can be seen as a categorification of a semisimple kk-algebra. For example, the group algebra of a finite group GG would categorify to the fusion category of GG-graded vector spaces. On the other hand, the notion of a 2-group is an internalisation of the notion of a group. A strict 2-group (which is essentially the same as a crossed module) in particular has an underlying group.

In this sense, a GG-crossed braided fusion category can be seen as some kind of categorification of a crossed module (G,H,δ:HG,α:GAut(H))(G, H, \delta\colon H \to G, \alpha\colon G \to Aut(H)), where the group HH is lifted to a fusion category, but GG is still a (finite) group. The boundary morphism δ\delta is replaced by a GG-grading, and the Peiffer rule (GG-graded commutativity) is categorified to a crossed braiding on 𝒞\mathcal{C}.

GG-crossed braided fusion category should also be related to monoidal bicategories.

Name

There are various names for this particular flavour of fusion category, involving permutations of the words “braided” and “crossed”, or possibly trading “braided” for “GG-braided”.

The latter choice has its justification in the fact that in general, it is not a braided category, but the braiding is in a sense twisted by the grading, just as the second group HH in a crossed module need not be abelian, but up to a group action.

Definition

A GG-crossed braided fusion category consists of the following:

  • A finite group GG,
  • a fusion category 𝒞\mathcal{C} over a field kk,
  • a GG-grading on 𝒞\mathcal{C},
  • a monoidal GG-action ρ\rho on 𝒞\mathcal{C} (i.e. a monoidal functor (ρ,ρ 2,ρ 0):G̲Aut (𝒞)(\rho, \rho^2, \rho^0)\colon \underline{G} \to Aut_\otimes(\mathcal{C}) from GG viewed as a discrete monoidal category to the category of tensor automorphisms of 𝒞\mathcal{C}) such that ρ(g 1)𝒞 g 2𝒞 g 1g 2g 1 1\rho(g_1)\mathcal{C}_{g_2} \subset \mathcal{C}_{g_1g_2g_1^{-1}},
  • for each gGg \in G, a natural isomorphism c g,X,Y:XYρ(g)(Y)Xc_{g,X,Y}\colon X \otimes Y \to \rho(g)(Y) \otimes X, where X𝒞 g,Y𝒞X \in \mathcal{C}_g, Y \in \mathcal{C}

satisfying

  • a GG-graded hexagon equation for cc:
(XY)Z α X,Y,Z c X,Y1 Z X(YZ) (ρ(g)(Y)X)Z c X,YZ α ρ(g)(Y),X,Z ρ(g)(YZ)X ρ(g)(Y)(XZ) ρ 2(g) Y,Z1 X 1 ρ(g)(Y)c X,Z (ρ(g)(Y)ρ(g)(Z))X α ρ(g)(Y),ρ(g)(Z),X ρ(g)(Y)(ρ(g)(Z)X) \array{ & & (X \otimes Y) \otimes Z \\ & \mathllap{{}^{\alpha_{X,Y,Z}}\swarrow} & & \mathrlap{\searrow^{c_{X,Y} \otimes 1_Z}} \\ X \otimes (Y \otimes Z) & & & & (\rho(g)(Y) \otimes X) \otimes Z \\ {}^{c_{X,Y \otimes Z}}\downarrow & & & & \downarrow^{\alpha_{\rho(g)(Y), X, Z}} \\ \rho(g)(Y \otimes Z) \otimes X & & & & \rho(g)(Y) \otimes (X \otimes Z) \\ {}^{\rho^2(g)_{Y, Z} \otimes 1_X}\downarrow & & & & \downarrow^{1_{\rho(g)(Y)} \otimes c_{X, Z}} \\ (\rho(g)(Y) \otimes \rho(g)(Z)) \otimes X & & \mathclap{\underset{\alpha_{\rho(g)(Y), \rho(g)(Z), X}}{\longrightarrow}} & & \rho(g)(Y) \otimes (\rho(g)(Z) \otimes X) }
  • compatibility of the crossed braiding and the group action.

TODO Commutative diagrams

If 𝒞\mathcal{C} carries extra structure (e.g. a pivotal structure), the group action is typically required to preserve it.

Examples

  • Every braided fusion category can be trivially graded, with the trivial action.
  • Every crossed module (G,H,δ,α)(G, H, \delta, \alpha) makes HH-graded vector spaces into a GG-crossed category. (Check/reference!)

(De-)Equivariantisation

See also equivariantisation for more details.

Let GG be a finite group, and 𝒞\mathcal{C} a braided fusion category.

A braided action of the (finite dimensional) representations Rep G\operatorname{Rep}_G on 𝒞\mathcal{C} is the same as an inclusion of Rep G\operatorname{Rep}_G in the symmetric centre 𝒞\mathcal{C}'. By deequivariantisation, this is basically the same as a braided GG-action on 𝒞 G\mathcal{C}_G (the category of internal k[G]k[G]-modules). This makes 𝒞 G\mathcal{C}_G into a GG-crossed braided fusion category, but most examples don’t arise like this, e.g. the grading obtained this way will always be trivial.

The idea is then to generalise the full inclusion Rep G𝒞𝒞\operatorname{Rep}_G \hookrightarrow \mathcal{C}' \hookrightarrow \mathcal{C} to a full inclusion Rep G𝒞\operatorname{Rep}_G \hookrightarrow \mathcal{C} that need not factor over 𝒞\mathcal{C}'. One can still deequivariantise the underlying fusion category of 𝒞\mathcal{C} with respect to the Rep G\operatorname{Rep}_G-action, but the procedure will not respect the braiding. Naturally, the result is not a braided fusion category, but a GG-crossed braided fusion category.

Vice versa, given a GG-crossed braided fusion category 𝒞\mathcal{C}, one can equivariantise it to a braided fusion category with a full inclusion of Rep G\operatorname{Rep}_G. When cc is the crossed braiding of 𝒞\mathcal{C}, the braiding of two equivariant objects (Xob𝒞,u g:ρ(g)(X)X)(X \in \operatorname{ob} \mathcal{C}, u_g\colon \rho(g)(X) \to X) and (Yob𝒞,v g:ρ(g)(Y)Y)(Y \in \operatorname{ob} \mathcal{C}, v_g\colon \rho(g)(Y) \to Y) is given by XYc X,Yρ(g)YXv g1 XYXX \otimes Y \xrightarrow{c_{X,Y}} \rho(g)Y \otimes X \xrightarrow{v_g \otimes 1_X} Y \otimes X.

4d extended TQFTs

TODO - Cui’s thesis

Higher Categories

TODO Cui’s 2-category

References

  • Section 4.4 in On braided fusion categories I, Drinfeld, Gelaki, Nikshych, Ostrik. ArXiv
  • Turaev
  • Müger
  • Cui

Last revised on January 26, 2018 at 06:08:31. See the history of this page for a list of all contributions to it.