Extending to a model structure is not a first-order property

TL;DR

This paper demonstrates that the existence of a model structure on a finitely bicomplete category with a specified subcategory of weak equivalences cannot be expressed in first-order logic, highlighting limitations in formal expressibility.

Contribution

It proves the non-first-order expressibility of model structures and characterizes all such structures when the category is a partial order.

Findings

Existence of model structures is not first-order expressible.

Characterization of model structures on partial orders.

Model structures are determined by their homotopy categories.

Abstract

Let $\mathcal{C}$ be a finitely bicomplete category and $\mathcal{W}$ a subcategory. We prove that the existence of a model structure on $\mathcal{C}$ with $\mathcal{W}$ as subcategory of weak equivalence is not first order expressible. Along the way we characterize all model structures where $\mathcal{C}$ is a partial order and show that these are determined by the homotopy categories.