nLab G-crossed braided fusion category

Intuition

Fusion categories over a field $k$ may be regarded as a categorification of semisimple $k$-algebras. 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 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, $G$-crossed braided fusion categories may be seen as some kind of categorification of crossed modules $\big(G, H, \delta\colon H \to G, \alpha\colon G \to Aut(H)\big)$, 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 categories can also be turned into monoidal bicategories.

Remark

(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 “$G$-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

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

Remark

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

Examples

Example

Every braided fusion category can be trivially graded, with the trivial action.

Example

Every crossed module $(G, H, \delta, \alpha)$ makes $H$-graded vector spaces into a $G$-crossed category. (Check/reference!)

Properties

(De-)Equivariantisation

See also equivariantisation for more details.

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 $X \otimes Y \xrightarrow{c_{X,Y}} \rho(g)Y \otimes X \xrightarrow{v_g \otimes 1_X} Y \otimes X$.

Relation to 4d extended TQFTs

TODO – Cui 2019

Relation to Higher Categories

TODO – Cui 2019‘s 2-category

References

• Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, Section 4.4 in: On braided fusion categories I, Selecta Mathematica. New Series 16 1 (2010) 1–119 [arXiv:0906.0620, doi:10.1007/s00029-010-0017-z]

• Turaev

• Müger

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

• Shawn X. Cui, Higher Categories and Topological Quantum Field Theories, Quantum Topology 10 4 (2019) 593-676 [arXiv:1610.07628, doi:10.4171/QT/128]

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

• Corey Jones, David Penneys, David Reutter, A 3-categorical perspective on $G$-crossed braided categories, J. London Math. Soc. 107 1 (2023) 333-406 [arXiv:2009.00405, doi:10.1112/jlms.12687]