On normal tensor functors and coset decompositions for fusion categories

We introduce the notion of double cosets relative to two fusion subcategories of a fusion category. Given a tensor functor $F : \C \to \D$ between fusion categories, we introduce an equivalence relation $\approx^F$ on the set $\Lambda_\C$ of isomorphism classes of simple objects of $\C$, and when $F$ is dominant, an equivalence relation $\approx_F$ on $\Lambda_\D$. We show that the equivalent classes of $\approx^F$ are cosets. We also give a description of the image of $F$ when it is a normal tensor functor, and we show that $F$ is normal if and only if the images of $\approx^F$ equivalent elements of $\Lambda_\C$ are colinear. We study the situation where the composition of two tensor functors $F=F'F"$ is normal, and we give a criterion of normality for $F"$, with an application to equivariantizations. Lastly, we introduce the radical of a fusion subcategory and compare it to its commutator in the case of a normal subcategory. We also give a description for the image of a normal tensor functor between any two fusion categories.