nLab G-crossed braided fusion category



Monoidal categories

Fusion categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



Fusion categories over a field kk may be regarded as a categorification of semisimple k k -algebras. 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 2-groups is an internalisation of the notion of groups. A strict 2-group (which is essentially the same as a crossed module) in particular has an underlying group.

In this sense, GG-crossed braided fusion categories may be seen as some kind of categorification of crossed modules (G,H,δ:HG,α:GAut(H))\big(G, H, \delta\colon H \to G, \alpha\colon G \to Aut(H)\big), 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 categories can also be turned into monoidal bicategories.


(Terminology) 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.



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}


  • 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.



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!)



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.

Relation to 4d extended TQFTs

TODO – Cui 2019

Relation to Higher Categories

TODO – Cui 2019‘s 2-category


Description of GG-crossed braided categories as monoidal bicategories and construction of a 4d TQFT:

Equivalence of certain Gray categories with GG-crossed braided categories and strictification:

Last revised on March 12, 2024 at 12:04:38. See the history of this page for a list of all contributions to it.