nLab graded fusion category

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

• Every fusion category has the trivial grading from the trivial group.
• $G$-graded vector spaces for a finite group $G$ are naturally a graded fusion category.
• The universal grading, see below.
• $G$-crossed braided fusion categories are a (kind of) categorification of crossed modules, and therefore carry a grading as part of the data.

Universal grading

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

• Shlomo Gelaki, Dmitri Nikshych, Nilpotent fusion categories
• Pavel Etingof, Dmitri Nikshych, Victor Ostrik, with an appendix by Ehud Meir, Fusion categories and homotopy theory
• Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, On braided fusion categories I