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Recall that the Strom-Model Structure is the model structure on compactly generated weakly Hausdorff spaces given by Hurewicz-cofibrations (having the homotopy extension property), Hurewicz-fibrations (having the homotopy lifting property) and homotopy equivalences. To verify this, one needs to show in particular that given a square $$\begin{array}{ccc} \;\;A & \longrightarrow & X\\ i\downarrow & & \;\;\downarrow p\\ \;\;B & \longrightarrow & Y \end{array}$$ with $i$ a Hurewicz-fibration and $p$ a Hurewicz-cofibration a lift $B \longrightarrow X$ exists, whenever $i$ or $p$ is a homotopy equivalence.

A proof of this can be found in May's More Concise Algebraic Topology Prop. 17.1.4, using the fact that Hurewicz-cofibrations are NDR-pairs (one of the many notions of neighborhood deformation retracts). But I wonder

Is there a more formal proof of this fact avoiding this explicit description of cofibrations, ie. using only properties of the interval $I=[0,1]$ as well as things like the $-\times X \dashv \operatorname{Map}(X,-)$ adjunction?

I am interested in this, because it would give a nice way to define Strom-like model structures on monoidal closed categories with a Berger-Moerdijk segment object. I am interested in such a general construction, because I feel like it would settle a previous question of mine and because I believe there to be at least two other examples of such a model structure (I don't want to say something wrong here. I still have to check minor details...).

It is immediate from the homotopy extension property (though took me some time to realize) that any cofibration, which is a homotopy equivalence, is a split monomorphism. But I failed to even show that such a retraction is homotopic to the homotopy inverse. Even if I could show formally that any Hurewicz-cofibration is a strong deformation retract, I wasn't able to deduce the desired lifting property.

So does anyone of you know a reference for this (or maybe on the other hand an obstruction making it impossible)?

Thank you very much, for your time and considerations.

Jonas Linssen
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    I'm going to go ahead and suggest that no such proof exists. The reason is that if you don't work with weak Hausdorff spaces, then the statement as written is not true, yet the interval in this larger category has the same formal properties you mention. Unless you have something else in mind, then the orthogonal characterisation cannot follow directly from only formal properties of the cofibrations in $Top$. – Tyrone Feb 03 '21 at 16:41
  • You mean, just taking compactly generated spaces and omitting weakly Hausdorff? I guess then cofibrations are closed Hurewicz cofibrations and I see where this is going, as we are still cartesian closed and have the interval... But is it just me thinking there ought to be a more categorical reason that things play out so nicely in the case of $\mathsf{CGWH}$-spaces? – Jonas Linssen Feb 03 '21 at 16:55
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    Strom's original proof is set in the category of all topological spaces. As a compact $T_2$ space $I$ still satisfies the adjunction $Top(X\times I,Y)=Top(X,Y^I)$ (the compact-open topology on $Y^I$ suffices). But, yeah, it suffices to work with CG spaces to get the idea. Another idea would be to consider the pointed category $CGWH_$, which is cartesian closed. The pointed cylinder has the same formal properties as elsewhere. You can give $CGWH_$ the slice model structure, but as far as I know it doesn't admit a model structure with pointed cofibrations/fibrations/homotopy equivalences. – Tyrone Feb 03 '21 at 17:11
  • Thank you very much, I should have noticed that. I would accept your comments as an answer, if you like. – Jonas Linssen Feb 03 '21 at 17:16
  • @Tyrone A pointed category is almost never cartesian closed. Do you mean monoidal closed? – Zhen Lin Feb 04 '21 at 01:29
  • @ZhenLin $CGWH_$ is closed, but yes, not cartesian closed. Thanks for the correction. The cylinder object in $CGWH_$ is the smash $X\wedge I_+$, however, and this is what I had in mind. – Tyrone Feb 04 '21 at 16:31

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