A model structure is left proper if the pushout of a weak equivalence along a cofibration is a weak equivalence. In the Hurewicz (or Strom) model structure on the category of topological spaces, weak equivalences are homotopy equivalences and cofibrations are defined in terms of a lifting property for Hurewicz fibrations. Is this model structure left proper?
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7$\begingroup$ Isn't it true that having all objects cofibrant implies left properness? $\endgroup$– David WhiteCommented May 5, 2019 at 2:52
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The answer is yes. I say this because I recall that the way you form homotopy pushout of a `prepushout' diagram $C\gets A\to B$ in the Hurewicz structure is
- map a diagram $\bar C \gets \bar A \to \bar B$ in which both arrows are (Hurewicz) cofibrations into the given one by a pointwise homotopy equivalence
- form the categorical pushout of the replacement diagram.
If $C\gets A$ is a homotopy equivalence, we can take the replacement to be $\bar A \xleftarrow{id} \bar A \to \bar B$.
Then the answer to the question is a result of the fact that the homotopy pushout in the Hurewicz structure is well-defined up to homotopy equivalence.
answered May 5, 2019 at 1:09