Is the category of metric spaces and continuous maps Quillen equivalent to Top?

Question

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.

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 06:16
Thank you. Tom's comment tells me that the answer to the question I asked, per se, is "no". — , 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), http://arxiv.org/abs/1106.3249 — Sergey Melikhov, Feb 06 '12 at 16:25

score 8 · Answer 1 · answered Jan 29 '12 at 15:28

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.

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? — , 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? — , Jan 30 '12 at 05:53

Is the category of metric spaces and continuous maps Quillen equivalent to Top?

1 Answers1

Linked