Skew-monoidal categories and the Catalan simplicial set

TL;DR

This paper introduces a new perspective on skew-monoidal categories using the Catalan simplicial set, showing it classifies skew-monoidal structures and clarifies their intrinsic form.

Contribution

It demonstrates that the Catalan simplicial set classifies skew-monoidal categories, providing an intrinsic justification for their structure and orientation.

Findings

Catalan simplicial set C describes skew-monoidal structures

Simplicial maps from C classify skew-monoidal categories

Extension to skew monoidales in monoidal bicategories

Abstract

The basic data for a skew-monoidal category are the same as for a monoidal category, except that the constraint morphisms are no longer required to be invertible. The constraints are given a specific orientation and satisfy Mac Lane's five axioms. Whilst recent applications justify the use of skew-monoidal structure, they do not give an intrinsic justification for the form the structure takes (the orientation of the constraints and the axioms that they satisfy). This paper provides a perspective on skew-monoidal structure which, amongst other things, makes it quite apparent why this particular choice is a natural one. To do this, we use the Catalan simplicial set C. It turns out to be quite easy to describe: it is the nerve of the monoidal poset (2, v, 0) and has a Catalan number of simplices at each dimension (hence the name). Our perspective is that C classifies skew-monoidal structures in the sense that simplicial maps from C into a suitably-defined nerve of Cat are precisely skew-monoidal categories. More generally, skew monoidales in a monoidal bicategory K are classified by maps from C into the simplicial nerve of K.