On monads and warpings

TL;DR

This paper explores the relationship between warpings on monoidal categories and pseudomonads on bicategories, extending the concepts to skew monoidal categories and introducing a normalization process.

Contribution

It establishes an equivalence between warpings and pseudomonads, and develops a normalization method for skew monoidal categories to ensure invertibility of the right unit map.

Findings

Warpings correspond to pseudomonads on one-object bicategories.

A normalization process produces a universal skew monoidal category.

The right unit map can be made invertible through normalization.

Abstract

We explain the sense in which a warping on a monoidal category is the same as a pseudomonad on the corresponding one-object bicategory, and we describe extensions of this to the setting of skew monoidal categories: these are a generalization of monoidal categories in which the associativity and unit maps are not required to be invertible. Our analysis leads us to describe a normalization process for skew monoidal categories, which produces a universal skew monoidal category for which the right unit map is invertible.