nLab $G$-crossed braided fusion category

Contents

Context

Monoidal categories

Fusion categories

monoidal categories

Contents

Intuition

A fusion category over a field $k$ can be seen as a categorification of a semisimple $k$-algebra. For example, the group algebra of a finite group $G$ would categorify to the fusion category of $G$-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 $G$-crossed braided fusion category can be seen as some kind of categorification of a crossed module $(G, H, \delta\colon H \to G, \alpha\colon G \to Aut(H))$, where the group $H$ is lifted to a fusion category, but $G$ is still a (finite) group. The boundary morphism $\delta$ is replaced by a $G$-grading, and the Peiffer rule ($G$-graded commutativity) is categorified to a crossed braiding on $\mathcal{C}$.

$G$-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 “$G$-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 $H$ in a crossed module need not be abelian, but up to a group action.

Definition

A $G$-crossed braided fusion category consists of the following:

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

satisfying

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

TODO Commutative diagrams

If $\mathcal{C}$ 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 $(G, H, \delta, \alpha)$ makes $H$-graded vector spaces into a $G$-crossed category. (Check/reference!)

(De-)Equivariantisation

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

A braided action of the (finite dimensional) representations $\operatorname{Rep}_G$ on $\mathcal{C}$ is the same as an inclusion of $\operatorname{Rep}_G$ in the symmetric centre $\mathcal{C}'$. By deequivariantisation, this is basically the same as a braided $G$-action on $\mathcal{C}_G$ (the category of internal $k[G]$-modules). This makes $\mathcal{C}_G$ into a $G$-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 $\operatorname{Rep}_G \hookrightarrow \mathcal{C}' \hookrightarrow \mathcal{C}$ to a full inclusion $\operatorname{Rep}_G \hookrightarrow \mathcal{C}$ that need not factor over $\mathcal{C}'$. One can still deequivariantise the underlying fusion category of $\mathcal{C}$ with respect to the $\operatorname{Rep}_G$-action, but the procedure will not respect the braiding. Naturally, the result is not a braided fusion category, but a $G$-crossed braided fusion category.

Vice versa, given a $G$-crossed braided fusion category $\mathcal{C}$, one can equivariantise it to a braided fusion category with a full inclusion of $\operatorname{Rep}_G$. When $c$ is the crossed braiding of $\mathcal{C}$, the braiding of two equivariant objects $(X \in \operatorname{ob} \mathcal{C}, u_g\colon \rho(g)(X) \to X)$ and $(Y \in \operatorname{ob} \mathcal{C}, v_g\colon \rho(g)(Y) \to Y)$ is given by $X \otimes Y \xrightarrow{c_{X,Y}} \rho(g)Y \otimes X \xrightarrow{v_g \otimes 1_X} Y \otimes X$.

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.