Monads of oplax actions are skew monoidales

TL;DR

This paper establishes that right skew monoidales can be characterized as monads of oplax actions within a bicategorical framework, providing a new perspective on their structure without restrictions on the ambient bicategory.

Contribution

It proves that right skew monoidales are equivalent to monads of oplax actions, using simplicial objects and the Catalan simplicial set, generalizing previous characterizations.

Findings

Right skew monoidales are monads of oplax actions.

A simplicial object in Cat models oplax actions as bicategories.

Bijective correspondence between monads of oplax actions and right skew monoidales.

Abstract

Szlach\'anyi showed that bialgebroids can be characterised using skew monoidal categories. The characterisation reduces the amount of data, structure, and properties required to define them. Lack and Street provide a bicategorical account of that same fact; they characterise quantum categories in terms of skew monoidal structures internal to a monoidal bicategory. A quantum category is an opmonoidal monad on an enveloping monoidale $R^\circ\otimes R$ in a monoidal bicategory. In a previous paper, we characterised opmonoidal arrows on enveloping monoidales as a simpler structure called oplax action. This is the second paper based on the author's PhD thesis. Here, motivated by the fact that opmonoidal monads are monads in the bicategory of monoidales, opmonoidal arrows, and opmonoidal cells; we prove that right skew monoidales are "monads of oplax actions". To do so, we arrange oplax actions as the 1-simplices of a simplicial object in $\mathsf{Cat}$. In nice cases this simplicial object is ought to be thought as a bicategory whose arrows are oplax actions, that is to say, it is weakly equivalent to a nerve of a bicategory. We define monads of oplax actions as simplicial maps out of the Catalan simplicial set and prove that these are in bijective correspondence with right skew monoidales whose unit has a right adjoint, no assumptions required on the ambient monoidal bicategory.