On the linear independency of monoidal natural transformations

TL;DR

This paper proves that monoidal natural transformations are linearly independent within the space of all natural transformations, leading to finiteness results for automorphisms and pivotal structures in finite tensor categories.

Contribution

It establishes the linear independence of monoidal natural transformations and derives finiteness results for automorphisms and pivotal structures in finite tensor categories.

Findings

Monoidal natural transformations are linearly independent within natural transformations.

The group of monoidal natural automorphisms on the identity functor is finite.

The set of pivotal structures on a finite tensor category is finite.

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.