Is it an open problem if $\mathbf{TOP}$ with Hurewicz (Strøm) model structure is cofibrantly generated?
2026-04-13 02:52:06.1776048726
Hurewicz model structure and cofibrantly generated model categories
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I have a vague memory of being told that someone has proven that the Strom model structure is not cofibrantly generated. I would ask Boris Chorny or Carles Casacuberta.
That said, a map $f \colon X \to Y$ is a Hurewicz fibration if and only if it lifts against the inclusion $Nf \to Nf \times I$, where $Nf$ denotes its mapping cocylinder. From this observation, there is a generalized (algebraic) version of the small object argument that allows one to formally construct the model-theoretic functorial factorization. This is explained in a paper "On the construction of functorial factorizations for model categories" that I wrote with Tobias Barthel.