





It's ugly to think of topological spaces as $[(X,d)]$ (or, rather, $(X,[d])$): it's better to use the simpler and more expressive abstraction $(X,\tau)$.

So, the question is:

  • Is there some kind of "homotopology" $h$ (whatever it is) that one can put on sets $S$ so that each homotopy type $[(X,\tau)]$ is fully described by a "homotopical space" (whatever it is) $(S,h)$ and homotopy classes of morphisms $[f]$ correspond to "$h$-compatible" (whatever it means) set-theoretic maps $F:S \rightarrow S'$?


  • $\begingroup$ Just out of curiosity: did the guy who voted to close the question regarded it as "no longer relevant" or as "offtopic"? $\endgroup$
    – Qfwfq
    Apr 17 '10 at 16:08
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    $\begingroup$ Well, in the light of the article linked by QiaochuYuan, I'd say it definitely is a question: "is the homotopy category of topological spaces a concrete category?". Perhaps I should edit to make it clear. $\endgroup$
    – Qfwfq
    Apr 18 '10 at 4:01
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    $\begingroup$ My understanding is that it is preferable to say "concretizable" rather than "a concrete category": one should always think of a concrete category as a category equipped with a distinguished forgetful functor to Set, since the same category can be concretized in different ways. $\endgroup$ Apr 18 '10 at 4:34
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    – Qfwfq
    Apr 18 '10 at 19:39


  • $\begingroup$ Perfect! I've accepted this answer. It's surprising that such a theorem ha sonly been proven in 2004. $\endgroup$
    – Qfwfq
    Apr 17 '10 at 15:35
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    $\begingroup$ I've never seen such a short and effective answer. It does not just answer the question, but along the way makes it precise in terms of category theory. +1. By the way, I think it's more easy to understand the proof that the homotopy category of chain complexes is not concrete. $\endgroup$ Apr 17 '10 at 15:41
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    $\begingroup$ Ouh, I've read this paper some years ago, but I thought it's a well-known fact. Can anybody provide historical references who proved this first? $\endgroup$ Apr 17 '10 at 15:45
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    $\begingroup$ @unknown: in fact it was proven by 1970, and republished in 2004. $\endgroup$ Apr 17 '10 at 15:47
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