On the linear independency of monoidal natural transformations

Abstract

Let $F, G: \mathcal{I} \to \mathcal{C}$ be strong monoidal functors from a skeletally small monoidal category $\mathcal{I}$ to a tensor category $\mathcal{C}$ over an algebraically closed field $k$. The set $Nat(F, G)$ of natural transformations $F \to G$ is naturally a vector space over $k$. We show that the set $Nat_\otimes(F, G)$ of monoidal natural transformations $F \to G$ 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.