On endomorphism algebras of functors with non-compact domain

TL;DR

This paper constructs VN-bialgebras in vector spaces from certain functors without requiring the usual compactness assumptions, expanding the scope of algebraic structures associated with monoidal categories.

Contribution

It introduces a new method to build VN-bialgebras from split-semigroupal functors without compactness constraints on the domain category.

Findings

Constructed VN-bialgebras in Vect_k from non-compact domain functors

Extended the theory of VN-bialgebras beyond compactness assumptions

Provided examples illustrating the new construction

Abstract

As a development of [2] and [3], we construct a "VN-bialgebra" in Vect_k for each k-linear split-semigroupal functor from a suitable monoidal category C to Vect_k. The main aim here is to avoid the customary compactness assumptions on generators of the domain category C (cf. [3]). Please note that the VN-bialgebras in Vect_k defined here are not necessarily von Neumann regular as k-algebras in the usual sense, the prefix "VN-" coming from the Set-based case of von Neumann regular semigroups.