Minimal model structures

TL;DR

This paper establishes the existence and uniqueness of minimal Quillen model structures on locally presentable categories with specified cofibrations and weak equivalences, using advanced categorical techniques.

Contribution

It constructs minimal model structures without set-theoretic assumptions, generalizing to semi-model categories and describing equivalences via Cisinski localizers.

Findings

Existence of unique minimal model structures under broad conditions

Construction of minimal models with prescribed cofibrations and weak equivalences

Application of advanced categorical tools like fat small object argument

Abstract

We prove, without set theoretic assumptions, that every locally presentable category C endowed with a tractable cofibrantly generated class of cofibrations has a unique minimal (or left induced) Quillen model structure. More generally, for any set S of arrows in C we construct the minimal model structure on C with the prescribed cofibrations and making all the arrows of S weak equivalences. We describe its class of equivalences as the "smallest Cisinski localizer containing S". Our proof rely on a careful use of the fat small object argument and J.~Lurie's "good colimits" technology and on the author previous work on combinatorial weak model categories and semi-model categories. We also obtain similar results for left semi-model categories.