Diagrammatics for lax and Frobenius monoidal functors and weak morphism classifiers

TL;DR

This paper introduces a diagrammatic construction of categories that classify lax, oplax, and Frobenius lax monoidal functors, extending the diagrammatic approach to monoidal functor theory.

Contribution

It provides an elementary, diagrammatic method to construct categories representing lax and Frobenius lax monoidal functors, generalizing existing diagrammatics.

Findings

Constructs L(C) for strict monoidal categories C.

Provides variants for oplax and Frobenius lax functors.

Uses diagrams analogous to McCurdy's lax monoidal functor diagrammatics.

Abstract

The theory of 2-monads entails that, for a strict monoidal category C, there is a strict monoidal category L(C) such that strict monoidal functors from L(C) are precisely the lax monoidal functors from C. We give an elementary, diagrammatic, construction of L(C) and of its variants for oplax and Frobenius lax functors. The diagrams used are analogous to the diagrammatics for lax monoidal functors studied by McCurdy.