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
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.
(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.
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
TODO Commutative diagrams
If $\mathcal{C}$ 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 $(G, H, \delta, \alpha)$ makes $H$-graded vector spaces into a $G$-crossed category. (Check/reference!)
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 $X \otimes Y \xrightarrow{c_{X,Y}} \rho(g)Y \otimes X \xrightarrow{v_g \otimes 1_X} Y \otimes X$.
TODO – Cui 2019
TODO – Cui 2019‘s 2-category
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:
Equivalence of certain Gray categories with $G$-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.