On the linear independency of monoidal natural transformations

الأصل · EN

Let F, G: I → C be strong monoidal functors from a skeletally small monoidal category I to a tensor category C over an algebraically closed field k. The set Nat(F, G) of natural transformations F → G is naturally a vector space over k. We show that the set Nat⊗(F, G) of monoidal natural transformations F → G is linearly independent as a subset of Nat(F, G). As a corollary, we can show that the group of monoidal natural automorphisms on the identity functor on a finite tensor category is finite. We can also show that the set of pivotal structures on a finite tensor category is finite.