On endomorphism algebras of separable monoidal functors

TL;DR

This paper demonstrates that the (co)endomorphism algebra of a sufficiently separable fibre functor into vector spaces over a field of characteristic zero forms a unital von Neumann core, a structure weaker than a Hopf algebra.

Contribution

It introduces the concept of a unital von Neumann core in Vect_k and relates it to endomorphism algebras of separable fibre functors, expanding the understanding of algebraic structures in monoidal categories.

Findings

Endomorphism algebra forms a unital von Neumann core

Weaker than Hopf algebra in Vect_k

In Set, corresponds to a group

Abstract

We show that the (co)endomorphism algebra of a sufficiently separable "fibre" functor into Vect_k, for k a field of characteristic 0, has the structure of what we call a "unital" von Neumann core in Vect_k. For Vect_k, this particular notion of algebra is weaker than that of a Hopf algebra, although the corresponding concept in Set is again that of a group.