Homotopical algebra is not concrete

TL;DR

This paper extends Freyd's result by providing a general method to demonstrate that the homotopy category of certain model categories cannot be concrete, deepening the understanding of the relationship between set theory and homotopy theory.

Contribution

It introduces a general approach to show non-concreteness of homotopy categories under specific conditions, advancing the theoretical understanding of homotopical algebra.

Findings

Homotopy categories of certain model categories are not concrete.

A general method to prove non-concreteness is developed.

The work links set theory with abstract homotopy theory.

Abstract

We generalize Freyd's well-known result that "homotopy is not concrete", offering a general method to show that under certain assumptions on a model category $\mathcal M$, its homotopy category $\text{ho}(\mathcal M)$ cannot be concrete. This result is part of an attempt to understand more deeply the relation between set theory and abstract homotopy theory.