# How many model category structures are there on Top?

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.

• 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. May 18, 2012 at 22:10
• May I ask why are you interested in counting such things? May 18, 2012 at 22:18
• According to ncatlab.org/nlab/show/…, Bousfield localization of the Serre model structure yields a plethora of new model structures on spaces. May 18, 2012 at 22:19
• 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. May 18, 2012 at 22:29
• @JeremyLane I was really curious May 19, 2012 at 14:05

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.