Model structures on finite total orders

TL;DR

This paper explores model structures on finite total orders, revealing a rich combinatorial framework linked to Catalan triangles, and extends previous operad theory to new structural insights.

Contribution

It systematically classifies all model structures on finite total orders and connects them to combinatorial objects like Catalan triangles, advancing homotopical combinatorics.

Findings

Enumeration of all model structures on finite total orders

Identification of Catalan triangle patterns in the structure

Extension of operad theory to new categorical structures

Abstract

We initiate the study of model structures on (categories induced by) lattice posets, a subject we dub homotopical combinatorics. In the case of a finite total order $[n]$, we enumerate all model structures, exhibiting a rich combinatorial structure encoded by Shapiro's Catalan triangle. This is an application of previous work of the authors on the theory of $N_\infty$-operads for cyclic groups of prime power order, along with new structural insights concerning extending choices of certain model structures on subcategories of $[n]$.