Triangulations, orientals, and skew monoidal categories

TL;DR

This paper constructs a concrete model of the free skew-monoidal category, linking it to triangulations, orientals, and Tamari lattices, and proves a coherence theorem that solves the word problem for skew monoidal categories.

Contribution

It provides the first explicit model of the free skew-monoidal category on a single object, connecting it to combinatorial structures and establishing a coherence theorem.

Findings

Constructed a concrete model of the free skew-monoidal category.

Established a coherence theorem with a faithful functor to ordinal-based categories.

Solved the word problem for skew monoidal categories.

Abstract

A concrete model of the free skew-monoidal category Fsk on a single generating object is obtained. The situation is clubbable in the sense of G.M. Kelly, so this allows a description of the free skew-monoidal category on any category. As the objects of Fsk are meaningfully bracketed words in the skew unit I and the generating object X, it is necessary to examine bracketings and to find the appropriate kinds of morphisms between them. This leads us to relationships between triangulations of polygons, the Tamari lattice, left and right bracketing functions, and the orientals. A consequence of our description of Fsk is a coherence theorem asserting the existence of a strictly structure-preserving faithful functor from Fsk to the skew-monoidal category of finite non-empty ordinals and first-element-and-order-preserving functions. This in turn provides a complete solution to the word problem for skew monoidal categories.