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.

Answer 1

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.