Fusion categories can be seen as generalisations of the representation category of finite groups. I'm interested in spherical fusion categories. I'm trying to find "interesting" functors from a spherical fusion category to a ribbon fusion category. Interesting means, that they should differ from those functors obtained from group homomorphisms in the following way:


Let $\phi: H \hookrightarrow G$ be an inclusion of finite groups. Define the functor $\phi^\*: \rho \mapsto \rho \circ \phi$ on the representation categories. Denote the regular representation by $\Omega\_G$ or $\Omega\_H$. Then $\Omega\_H$ is a subobject of $\phi^\* \Omega\_G$, more specifically, $\phi^\* \Omega\_G \cong \oplus^n \Omega\_H$, with $n = \lvert G/H \rvert$ the number of cosets.


*Are there pivotal functors $F: \mathcal{C} \to \mathcal{D}$ where $\mathcal{C}$ is spherical fusion and $\mathcal{D}$ is ribbon fusion, such that $\Omega\_\mathcal{D}$ is a subobject of $F\Omega\_\mathcal{C}$, but $F\Omega\_\mathcal{C}$ is **not** isomorphic to $\oplus^n \Omega\_\mathcal{D}$?*


$\Omega\_\mathcal{C}$ denotes the sum over all the simple objects in $\mathcal{C}$.


(This is related to my earlier question here: [Is the category of spherical fusion categories regular? (i.e. is image factorisation possible?)](https://mathoverflow.net/questions/160677/is-the-category-of-spherical-fusion-categories-regular-i-e-is-image-factorisa))



Pivotal functors of that are substantially different from finite group homomorphisms
I've been thinking about this since longer already and just realised a really easy example. I was a bit of a blockhead in thinking that for any inclusion functor $F$, we must have that $F\Omega\_\mathcal{C}$ is a subobject of $\Omega\_\mathcal{D}$, which is not true.


Consider the category of $U\_qSU(2)$-tilting modules modulo the negligible ones at a root of unity. Its simple objects are labelled by spins $j \in \left\{0, \frac{1}{2}, 1,\ldots{}, r\right\}$. Consider the subcategory spanned by the objects $\left\{0, \frac{1}{2} \oplus \frac{1}{2}, 1 \oplus 1,\ldots{}, r \oplus r\right\}$. It's closed under the monoidal product, so we have a monoidal subcategory. Now the sum of all simple objects in the subcategory clearly contains all simple objects of the whole category, but with different multiplicities, the monoidal identity only once and the rest twice.


