Abstract
We introduce the notion of double cosets relative to two fusion subcategories of a fusion category. Given a tensor functor \(F : {\mathcal {C}} \to {\mathcal {D}}\) between fusion categories, we introduce an equivalence relation ≈ F on the set \(\Lambda _{\mathcal {C}}\) of isomorphism classes of simple objects of \({\mathcal {C}}\), and when F is dominant, an equivalence relation ≈ F on \(\Lambda _{\mathcal {D}}\). We show that the equivalence classes of ≈ F are cosets. We also give a description of the image of F when F is a normal tensor functor, and we show that F is normal if and only if the images of ≈ F equivalent elements of \(\Lambda _{\mathcal {C}}\) 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.