Composition closed premodel structures and the Kreweras lattice

TL;DR

This paper explores the combinatorial structure of composition-closed premodel structures on finite lattices, revealing connections to the Kreweras lattice and noncrossing partitions, and characterizing model structures via tricolored trees.

Contribution

It introduces a natural refinement of the inclusion order of weak factorization systems that detects composition-closed premodel structures, linking lattice theory with combinatorial structures.

Findings

Intervals detect composition-closed premodel structures

Kreweras lattice arises as a refinement for total orders

Model structures correspond to certain tricolored trees

Abstract

We investigate the rich combinatorial structure of premodel structures on finite lattices whose weak equivalences are closed under composition. We prove that there is a natural refinement of the inclusion order of weak factorization systems so that the intervals detect these composition closed premodel structures. In the case that the lattice in question is a finite total order, this natural order retrieves the Kreweras lattice of noncrossing partitions as a refinement of the Tamari lattice, and model structures can be identified with certain tricolored trees.