This question is related to Pivotal functors of that are substantially different from finite group homomorphisms. A tensor functor F:C→D is called dominant (sometimes called “surjective”) if for any Y:D, there is an X:C such that Y is a subobject of FX. It is known (“On fusion categories” by Pavel Etingof, Dmitri Nikshych, and Viktor … Read more
The Nichols-Richmond theorem is a result on cosemisimple Hopf algebras, proved in their paper. It was restated for integral fusion categories by Dong-Natale-Vendramin (Theorem 3.4 here): Theorem: Let $x \in {\small \rm Irr}(\mathcal{C})$ with ${\small \rm FPdim} \ x = 2$. Then at least one of the following holds: $G[x] \neq 1.$ $\mathcal{C}$ has a … Read more
We refer to the book Tensor Categories (by Etingof-Gelaki-Nikshych-Ostrik) for all the notions mentioned in this post. A fusion category is skeletal if two isomorphic objects are always equal. Every fusion category is equivalent to a skeletal one (i.e. a skeleton, by choosing one object in each isomorphism class). A fusion category is strict if … Read more
Suppose that C is a ribbon monoidal category with dominant ribbon functors F_1: C->D_1 and F_2: C->D_2 such that D_1 and D_2 are modular tensor categories, does it follow that D_1 and D_2 are equivalent as MTCs? Here dominant means that every object in the target is a summand of an object in the image … Read more
This question is inspired by this paper of Scott Morrison and Kevin Walker. Consider a fusion category C of rank r, and its Drinfeld center Z(C) of rank s. Let Ni=(nkij), 1≤i≤r, be the r×r fusion matrices of C. Let IC=(mkl) be the r×s matrix for the induction from C to Z(C). Consider c(C,k,a)=card({(i,j) | nkij=a}) and … Read more
We refer to Chapter 8 of the book Tensor Categories for notions related to modular tensor categories and J.P. Serre for the basic theory of linear representations of finite groups over C. Let G be a finite group, VecωG be a category of finite dimensional G-graded vector spaces (potentially twisted by some non-trivial 3-cocycle ω) … Read more
Consider the following fusion categorie F(i) with integer parameter i. Simple objects are 1,a,A,B (where a and A are conjugates). Nontrivial fusion rules are a⨂a=A (and conjugate), B⨂a=B (and conjugate), B⨂B=1⨁a⨁A⨁i∗B (i.e. multiplicity i). The system is well known for i=2 (all quantum dimensions are integer then – offhand I can’t give the reference but … Read more
Multi-fusion categories are a generalization of fusion categories with a non-simple unit. The direct sum of two multi-fusion categories is again a multi-fusion category. By irreducible I mean that a multi-fusion category cannot be written as a non-trivial direct sum. Are all such irreducible multi-fusion categories fusion categories? Bonus question: Are there multi-fusion categories that … Read more
Given a category C, we can define the category of endofunctors Cat(C), with objects functors F:C→C and morphisms natural transformations. Since Cat is a 2-category, Cat(C) is naturally endowed with a strict monoidal product, which is functor composition. I’m interested in additional structure on such categories. Notice that if we set C=Vectfin.dim. and restrict to … Read more
Every self-dual simple object $X$ in a fusion category can canonically be assigned a number $a$, from its “snake” associator element: The square of $a$ equals Muger’s “squared dimension” of $X$, an invariant of the fusion category. Let us work over $\mathbb{C}$. Then $a^2$ is positive real number, as shown by Etingof, Nikshych, and Ostrik. … Read more
