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I am looking for models of ${\mathsf{Top}}$ distinct from modifications of simplicial sets. The above question should be understandable to the reader. I'll add more details when I get access to a proper computer. Let ${\mathsf{Met}}$ be the category of metric spaces and continuous maps. Then there is an embedding ${\mathsf{Met}}\hookrightarrow {\mathsf{Top}}$. Is this embedding a Quillen equivalence?

Edit: Professor May explains below that my question is not precise, per se, as there is more than one Quillen inequivalent model categorial structures on ${\mathsf{Top}}$.

Edit: The answer is no. Tom commented that the category ${\mathsf{Met}}$ does not have all small colimits. Thus the embedding ${\mathsf{Met}}\hookrightarrow {\mathsf{Top}}$ cannot be a Quillen equivalence.

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    $\begingroup$ Lots of diagrams in this category don't have colimits. $\endgroup$
    – Tom Goodwillie
    Commented Jan 29, 2012 at 5:38
  • $\begingroup$ So maybe we should ask for the $\infty$-category which we obtain by localizing the category in question at weak equivalences. $\endgroup$
    – Thomas Nikolaus
    Commented Jan 29, 2012 at 6:16
  • $\begingroup$ Say by using a relative category. $\endgroup$
    – David Roberts ♦
    Commented Jan 29, 2012 at 6:28
  • $\begingroup$ Thank you. Tom's comment tells me that the answer to the question I asked, per se, is "no". $\endgroup$
    – user2529
    Commented Jan 29, 2012 at 12:58
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    $\begingroup$ The category of metric spaces and uniformly continuous maps has many finite colimits - not all, but enough for the purposes of Baues' cofibration category (or Brown's category of cofibrant objects), arxiv.org/abs/1106.3249 $\endgroup$
    – Sergey Melikhov
    Commented Feb 6, 2012 at 16:25

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The phrasing of your question prompts me to emphasize a model categorical difference between spaces and simplicial sets. With the standard weak equivalences, there is just one standard model structure on simplicial sets. But with spaces, there is a natural trichotomy of interrelated model structures, two of which are Quillen equivalent to each other and to simplicial sets. There is an h-model structure with actual homotopy equivalences as weak equivalences and with Hurewicz (or h) cofibrations and fibrations. There is a q-model structure with weak homotopy equivalences as weak equivalences, Serre fibrations as q-fibrations, and retracts of relative cell complexes as q-cofibrations. And there is a mixed (or m) model structure with the q-equivalences and h-fibrations as the m-equivalences and m-fibrations. The m-cofibrant objects are the spaces of the homotopy types of CW-complexes, and algebraic topology over most of its history has implicitly worked in the m-model structure. The trichotomy carries over to chain complexes. A recent exposition is in the book ``More concise algebraic topology: localization, completion, and model categories'' by Kate Ponto and myself.

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    $\begingroup$ Professor May, which two of the model structures on spaces are Quillen equivalent to each other and to simplicial sets? Is the last model structure Quillen inequivalent to the rest? $\endgroup$
    – user2529
    Commented Jan 29, 2012 at 15:57
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    $\begingroup$ @Colin: The m-model structure and the q-model structure are Quillen equivalent to the standard model structure on simplicial sets. $\endgroup$
    – Dmitri Pavlov
    Commented Jan 29, 2012 at 17:47
  • $\begingroup$ @Dmitri: As you explained, simplicial sets are an algebraic model for the m-model and q-model structures. In the same vein, is there an algebraic model for the h-model structure? $\endgroup$
    – user2529
    Commented Jan 30, 2012 at 5:53

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