On the linear independency of monoidal natural transformations

Abstract:

Let $F$, $G: \mathcal {I} \to \mathcal {C}$ be monoidal functors from a monoidal category $\mathcal {I}$ to a linear abelian rigid monoidal category $\mathcal {C}$ over an algebraically closed field $\mathbf {k}$. Then the set $\mathrm {Nat}(F, G)$ of natural transformations $F \to G$ is naturally a vector space over $\mathbf {k}$. Under certain assumptions, we show that the set of monoidal natural transformations $F \to G$ is linearly independent as a subset of $\mathrm {Nat}(F, G)$.

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.