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Do the classes of pointed Hurewicz cofibrations, pointed Hurewicz fibrations and pointed homotopy equivalences give a model structure on pointed (compactly generated weak Hausdorff) topological spaces that is compatible with the smash product?

Maybe, this would work with another class of fibrations? (I am interested such a monoidal model structure with these precise classes of cofibrations and weak equivalences).

A reference is welcome!

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Yes. According to the nLab page for the Strom model structure, this is proven in Section 6.4 of May's Concise Course in Algebraic Topology. The point is that a closed inclusion is a cofibration if and only if it's an NDR pair, and you can use that to check that the pushout corner map is a (trivial) cofibration as required. Also, Donald Yau and I made use of this fact in Example 4.5 of this paper.

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  • $\begingroup$ Is that really true? I know that the category of pointed spaces inherits a monoidal model structure from the Hurewicz model structure on unbased spaces. But in this model structure, cofibrations are unbased Hurewicz cofibrations that are also based maps instead of the based cofibrations I want to look at. [I know that they coincide for well pointed spaces (that is shown in Concise), but I do not think they do in general.] $\endgroup$
    – user09127
    Commented Sep 24, 2018 at 20:46
  • $\begingroup$ All three of the links above are for the unpointed situation. If I find time, I can think about the pointed situation, but I'll just be trying to check if the argument in Concise goes through for your types of cofibrations. $\endgroup$
    – David White
    Commented Sep 24, 2018 at 22:49
  • $\begingroup$ Have you seen this other mathoverflow question about the pointed Strom model structure? mathoverflow.net/questions/47756 It seems you must allow the weak equivalences to be non-pointed in order to get a model structure. When you restrict to the cofibrant objects (the well-pointed spaces, aka non-degenerately based spaces) then they become pointed homotopy equivalences. So, you have to work with the model structure I was using in my answer, inherited from unbased spaces. $\endgroup$
    – David White
    Commented Sep 24, 2018 at 23:28
  • $\begingroup$ I see. I'll accept your answer for the sake of finishing this question (even though that's your last comment that convinced me). It would nevertheless be interesting to know whether the last model structure you mention is monoidal! $\endgroup$
    – user09127
    Commented Sep 26, 2018 at 14:41

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