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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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Lots of diagrams in this category don't have colimits. – Tom Goodwillie Jan 29 '12 at 5:38
So maybe we should ask for the $\infty$-category which we obtain by localizing the category in question at weak equivalences. – Thomas Nikolaus Jan 29 '12 at 6:16
Say by using a relative category. – David Roberts Jan 29 '12 at 6:28
Thank you. Tom's comment tells me that the answer to the question I asked, per se, is "no". – user2529 Jan 29 '12 at 12:58
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), – Sergey Melikhov Feb 6 '12 at 16:25

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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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? – user2529 Jan 29 '12 at 15:57
@Colin: The m-model structure and the q-model structure are Quillen equivalent to the standard model structure on simplicial sets. – Dmitri Pavlov Jan 29 '12 at 17:47
@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? – user2529 Jan 30 '12 at 5:53

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