Contents

### Context

#### Monoidal categories

##### Fusion categories

monoidal categories

With braiding

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

## Definition

###### Definition

Let $\mathcal{C}$ be a semisimple linear(Vect-enriched) (over a field $k$) category and $G$ a finite group. A $G$-grading on $\mathcal{C}$ is a direct sum decomposition of $\mathcal{C}$ into homogeneous components $\mathcal{C}_g$, which are again $k$-linear semisimple categories:

$\mathcal{C} \simeq \bigoplus_{g \in G} \mathcal{C}_g$
###### Definition

A graded fusion category is a fusion category with a chosen $G$-grading for some group, such that the monoidal product maps $\mathcal{C}_g \times \mathcal{C}_h$ into $\mathcal{C}_{g h}$.

###### Alternative Definition

Let $\Lambda_\mathcal{C}$ be the set of equivalence classes of simple objects in $\mathcal{C}$. A $G$-grading is a map $\operatorname{deg}\colon \Lambda_\mathcal{C} \to G$ such that for any two simples $X$ and $Y$ and any simple subobject $Z \hookrightarrow X \otimes Y$, we have $\operatorname{deg}([Z]) = \operatorname{deg}([X]) \operatorname{deg}([Y])$.

## Terminology

• The trivial component $\mathcal{C}_1$ of the grading is a full fusion subcategory.
• A $G$-extension of a fusion category $\mathcal{D}$ is a $G$-graded fusion category $\mathcal{C}$ such that $\mathcal{D} \simeq \mathcal{C}_1$.
• A grading is faithful if $\mathcal{C}_g \neq 0$ for all $g \in G$. This is a standard assumption.
• The adjoint category $\mathcal{C}_{\text{ad}}$ is the full fusion subcategory of $\mathcal{C}$ spanned by objects of the form $X \otimes X^*$.

## Examples

From the definition, it is clear that gradings are covariant in the group. A group homomorphism $\phi\colon G \to H$ gives an obvious map of the set of gradings:

$\mathcal{C} \simeq \bigoplus_{h \in H} \bigoplus_{g \in \phi^{-1}(h)} \mathcal{C}_g$

or

$\operatorname{deg}_H = \phi \circ \operatorname{deg}_G$

Morphisms of gradings are therefore simply group homomorphisms.

For every fusion category $\mathcal{C}$, there exists a universal grading by a group $U_\mathcal{C}$. It has the following properties:

• It is faithful.
• The trivial component is the full fusion subcategory spanned by objects of the form $X \otimes X^*$.
• Every full fusion subcategory $\mathcal{D} \subset \mathcal{C}$ containing the adjoint category $\mathcal{C}_{\text{ad}}$ is of the form $\mathcal{D} \simeq \bigoplus_{h \in H} \mathcal{C}_h$ for some subgroup $H \subset U_\mathcal{C}$.
• The group of monoidal automorphisms of the identity functor is canonically isomorphic to $\operatorname{Hom}(U_\mathcal{C}, k^\times)$.

## References

Last revised on August 29, 2017 at 13:21:59. See the history of this page for a list of all contributions to it.