Skew monoidal monoids

TL;DR

This paper investigates skew monoidal monoids, revealing their structure via dual bialgebroids and exploring their module categories, providing new insights into their algebraic properties and potential Hopf structures.

Contribution

It introduces the concept of skew monoidal monoids, characterizes their associated bialgebroids, and analyzes their module categories and possible closed and Hopf structures.

Findings

Skew monoidal monoids have dual bialgebroids describing their symmetries.

These bialgebroids are submonoids of their base and rank 1 free over it.

The paper provides equivalent definitions and studies module categories and Hopf structures.

Abstract

Skew monoidal categories are monoidal categories with non-invertible `coherence' morphisms. As shown in a previous paper bialgebroids over a ring R can be characterized as the closed skew monoidal structures on the category Mod R in which the unit object is R. This offers a new approach to bialgebroids and Hopf algebroids. Little is known about skew monoidal structures on general categories. In the present paper we study the one-object case: skew monoidal monoids (SMM). We show that they possess a dual pair of bialgebroids describing the symmetries of the (co)module categories of the SMM. These bialgebroids are submonoids of their own base and are rank 1 free over the base on the source side. We give various equivalent definitions of SMM, study the structure of their (co)module categories and discuss the possible closed and Hopf structures on a SMM.