Coherence for Monoidal Endofunctors

TL;DR

This paper establishes coherence results for monoidal endofunctors in various monoidal categories, providing foundational insights that support the study of monoidal monads and comonads.

Contribution

It extends coherence results to monoidal endofunctors without symmetry and in categories with diagonals or finite products, advancing the theoretical framework.

Findings

Coherence results proved without symmetry

Extended coherence to categories with diagonals or finite products

Foundation for coherence of monoidal monads and comonads

Abstract

The goal of this paper is to prove coherence results with respect to relational graphs for monoidal endofunctors, i.e. endofunctors of a monoidal category that preserve the monoidal structure up to a natural transformation that need not be an isomorphism. These results are proved first in the absence of symmetry in the monoidal structure, and then with this symmetry. In the later parts of the paper the coherence results are extended to monoidal endofunctors in monoidal categories that have diagonal or codiagonal natural transformations, or where the monoidal structure is given by finite products or coproducts. Monoidal endofunctors are interesting because they stand behind monoidal monads and comonads, for which coherence will be proved in a sequel to this paper.