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Call a model structure on $Top$ (the category of topological spaces) standard, if the weak equivalences are the weak homotopy equivalences. In this nLab page, two standard model structures on $Top$ are defined. One is the Quillen model structure, in which fibrations are the Serre fibrations, and one is the Mixed model structure, in which the fibrations are the Hurewicz fibrations. Since every Hurewicz fibrations is a Serre fibration, The Mixed Model Structure has more cofibrations then the Quillen Model Structure. For e.g., the cofibrant objects in the Mixed model structure are the topological spaces that are homotopy equivalent to $CW$-complexes, while the cofibrant objects in the Quillen model structure are the topological spaces that are retracts of $CW$-complexes. My Question is the following:

Is there a standard model structure on $Top$ with more cofibrations then the Mixed model structure?

Edit: Here is a counterexample that can give a certain upper bound on the class of cofibrations in any standard model structure on $Top$.

Let $W$ denote the Warsaw circle. The Warsaw circle can be defined as the subspace of the plane $R^2$ consisting of the graph of $y = \sin(1/x)$, for $x\in(0,1]$, the segment $[−1,1]$ in the $y$ axis, and an arc connecting $(1,\sin(1))$ and $(0,0)$ (which is otherwise disjoint from the graph and the segment). It can be shown that $W$ is weakly contractible but not contractible (see here for more details).

Let $I$ denote the segment $[−1,1]$ in the $y$ axis. I claim that the embedding $I\to W$ cannot be a cofibration in any standard model structure on $Top$. Indeed, suppose that we are given a standard model structure on $Top$ such that $I\to W$ is a cofibration. Then $I\to W$ is an acyclic cofibration, since $W$ is weakly contractible. Now take the pushout of $I\to W$ along $I\to *$. Since acyclic cofibrations are closed under pushouts, and since $W/I\cong S^1$ is just the usual circle, we get that $*\to S^1$ is an acyclic cofibration. Contradiction.

Note that the Warsaw circle is a compact metrizable space (being a bounded closed subspace of $R^2$) and $I\to W$ is a closed embedding. In particular, we get that there is no standard model structure on $Top$ in which every closed embedding between compact metrizable spaces is a cofibration.

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Ilan Barnea
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  • Your question made the following thread pop into my mind. I have no idea if it'll be useful and won't have time to think about it till next week, but let me share while it's on my mind: http://mathoverflow.net/questions/100153 – David White Aug 04 '15 at 10:48
  • I don't think so because the conclusion there is that the left determined model structure and the Quillen model structure are equal. – Ilan Barnea Aug 04 '15 at 11:30
  • Here is a plan of attack. Use $\Delta$-generated spaces, produce a model structure there, and try to pass it via adjunction to Top. Rosicky and Tholen in "Left-determined model..." prove that under Vopenka in any locally presentable category with fixed $W$ (here weak homotopy equivalences) then for any set $I$ of maps there is a left determined model structure with cofibrations generated by $I$. Take $I$ to be $\emptyset \to \ast$ and you get monomorphisms, which is more cofibrations than the mixed model structure. – David White Aug 12 '15 at 12:51
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    Also, have you read Tibor Beke's work? He has a paper called "How (non)unique is the choice of cofibrations?" which you might enjoy. That paper is really about the setting of a combinatorial model category, but doesn't require Vopenka. There might be a way to rig up an example on $\Delta$-generated spaces with more cofibrations than the mixed model structure, but I could not see how to do it. Beke's approach requires a careful understanding of $Ex^\infty$ and so seems to require sSet rather than Top. – David White Aug 12 '15 at 12:55
  • @DavidWhite I think the result of Rosicky and Tholen says that the weak saturation of any set of $I$ of morphisms in a locally presentable category is the class of cofibrations of a unique left-determined model structure. So how can you fix $W$ in advance? – Ilan Barnea Aug 14 '15 at 01:07
  • You are right; you can't fix W. However, by the nature of the left-determined model structure you know you can obtain a model structure with the W you want (weak homotopy equivalences) as a left Bousfield localization of the left determined one. I mean, you've already assumed Vopenka, so this gets rid of the problem of needing to find a SET of maps to invert. – David White Aug 14 '15 at 13:25
  • @DavidWhite Thanks, but I still don't get something. Why would taking $I$ to be $\phi\to *$ produce all monomorphisms? To me it looks like producing only maps of the form $X\to X\coprod D$, for $D$ a discrete space. Even taking a larger set like all $\partial\Delta^n\to \Delta^n$ looks to me like producing the cofibrations in the Quillen model structure which are less then the cofibrations in the mixed model structure. How are we getting more cofibrations? – Ilan Barnea Aug 14 '15 at 17:54
  • I was about to write something about proving the existence of a set $I$ to generate the monomorphisms, but I realized a bigger problem. This paper might contain mistakes. Look at the comment thread in http://mathoverflow.net/questions/135135. Anyway, my comment was only meant as a sketch of an approach, not a full answer. I agree work would have to be done to turn the sketch into a real proof. Gaucher has thought a lot recently about left determined models and has a preprint on his homepage which might be more useful than my musings. – David White Aug 14 '15 at 19:16
  • Have you seen: J. Rosický and W. Tholen, Erratum to "Left-determined model categories and universal homotopy theories", Trans. Amer. Math. Soc. 360 (2008), 6179-6179. Maybe they fix everything here – Ilan Barnea Aug 15 '15 at 22:25
  • Just a remark: the cofibrant objects in the Quillen model structure on Top are not retracts of CW complexes as you wrote, but rather retracts of cell-complexes. – MatanP Sep 21 '15 at 15:15
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    Hi Matan. I was not aware there is a difference – Ilan Barnea Oct 01 '15 at 11:23

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