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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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I recently started learning a little model 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 weak 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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How many model category structures are there on Top?

I recently started learning a little model 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 equivalences, Hurewicz fibrations and weak 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.