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I recently started learning a little model category theory and in particular I found this nice exercise. I only know a little topology, but this prompted me to wonder how many model category structures there may be on Top. I am aware of three: Serre fibrations and weak homotopy equivalences, Hurewicz fibrations and homotopy equivalences, and the usual model category of rational homotopy theory [Dwyer and Spalinski]. A secondary question could be how many homotopy theories there are since it is known that the first two I mention give the same homotopy theory.

*This question is a little out of my league right now. I hope that is ok. I'm not even sure how difficult this question is.

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The model structure that uses weak homotopy equivalences and Hurewicz fibrations is called the "mixed model structure"––it is attributed to Cole. There is yet another model structure due to Strøm: it uses Hurewicz fibrations and homotopy equivalences. So there are at least four different ones. The Strøm structure and the Serre/Cole structures have distinct homotopy theories. –  John Klein May 18 '12 at 22:10
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May I ask why are you interested in counting such things? –  Fernando Muro May 18 '12 at 22:18
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According to ncatlab.org/nlab/show/…, Bousfield localization of the Serre model structure yields a plethora of new model structures on spaces. –  John Klein May 18 '12 at 22:19
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There's no mathematical content to the following comment, but I'll say it anyway: what you're learning isn't model theory, but model category theory. Model theory is something else entirely. –  Tom Leinster May 18 '12 at 22:29
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@JeremyLane I was really curious –  Fernando Muro May 19 '12 at 14:05

1 Answer 1

Let's work in simplicial sets, where things are generally nicer (in particular, the standard model structure is combinatorial). I can refer to simplicial sets as a model for topological spaces because sSet is Quillen equivalent to Top and because simplicial sets can all be realized as topological spaces. In this setting, there are infinitely many different model structures, and this is proven in Tibor Beke's "Fibrations of Simplicial Sets." These model structures are all homotopically equivalent (by which I mean there is a chain of Quillen equivalences between any two). The difference between them is a difference of cofibrations and fibrations. So far we have countably many different structures. On page 2, Beke comments that taking products of these model structures demonstrates that there is no upper bound on the cardinality of possible cofibration classes. In a combinatorial setting like sSet, if you restrict attention to set-generated cofibration classes then all will be homotopically equivalent. This idea gives you the potential for arbitrarily many model structures, but we'd have to check that they all have different fibrations (this is the heart of his paper above).

If you want non-equivalent model structures then that's even easier. I wrote an answer here explaining truncated model structures, where $f$ is a weak equivalence if $\pi_k(f)$ is for $n\leq k\leq m$). Varying $n$ and $m$ give you infinitely many non-equivalent model structures. The non-equivalence can be seen by looking at how two different ones see wedges of Eilenberg-Mac Lane spaces. Again, this gives countably many, but I feel like there should be more.

As John Klein comments above, Bousfield localization helps you construct non-equivalent model structures. The idea now is that you are leaving the cofibrations fixed and increasing the weak equivalences by formally inverting a chosen set of maps. As sSet is combinatorial, every model category so created will also be combinatorial, and will therefore admit Bousfield localization again. There are certainly a class worth of choices for sets of maps to invert, but not all choices will give different model structures. Another natural idea would be to look at localization of the homotopy category and ask whether all come from Bousfield localizations of the model category. Dror Farjoun conjectured that this is the case. Casacuberta and Smith proved this under the assumption of Vopenka's principle from set theory (it's known that the standard axioms of ZFC cannot imply Vopenka). Casacuberta has also written other papers in the same vein which might interest you. He has several with Rosicky and/or Chorny (who sometimes frequents math overflow). The ones featuring the Orthogonal Subcategory Problem are very nice, and suggest again that questions about localization are often related to questions in set theory. Incidentally, if someone has an argument that there should be even uncountably many different Bousfield localizations on sSet I'd be curious to hear it. Getting a class worth would be great too.

Final remark: if you really want to work with compactly generated spaces or spaces homeomorphic to CW complexes instead of sSet then the situation is harder because you're no longer combinatorial. My answer in the middle paragraph about truncated model structures still holds. Tibor Beke addresses the answer in paragraph one at the very end of his paper. For Bousfield localization you need to use cellularity rather than combinatoriality, but you'll again have a class worth of possible model structures and will have to determine which are the same (and now you won't have the results of Casacuberta and coauthors because Top is not combinatorial).

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