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Re #3: I doubt one can use the adverb “traditionally” to characterize the other approach. Quillen’s original book on model categories, which is as traditional as one could possibly get, defines the model structure on topological spaces by transferring it from simplicial sets. Only later other authors transferred it in the opposite direction.
I’m looking at section II.3 of Quillen’s book. The fact that topological spaces are a model category is Theorem 1. The fact that simplicial sets are a model category is Theorem 3. So I don’t think that the former model structure could be transferred from the latter. (Note that Quillen defines “simplicial model category” in section II.2, before he’s proven that simplicial sets are themselves a model category.)
My memory of Quillen (I do not have it in front of me) is that he intends the three classes of w.e., fib and cofib to have the ’traditional ’meaning (e.g. from J. H. C. Whitehead, etc. and Whitehead is not only ’traditional’ it is more or less ’classical’, unless you go pre-WWII;-)), but as Quillen define homotopy groups for simplicial sets via geometric realisation, which structure has precedence is slightly problematic. It is also slightly irrelevant. The singular complex predates Quillen by some years, and is the motivation for the definition of simplicial sets. It dates from 1950 and Eilenberg-Zilber’s Annals paper. I am wary of ’traditional’ and ’classical’ as adjectives to describe mathematical ideas although I must admit to using them myself in what I write.
I have a slight feeling of unease about the paragraph in this entry on weak equivalences as I find it a bit too complicated, but do not see what to do to simplify it … or I would try to do it myself. Perhaps reducing what is said here to saying just that weak equivalences in simplicial sets were initially defined via those in topological spaces, might clear up the issue. (The construction using combinatorial methods is introduced by Kan (1958) as being secondary’).
In fact Quillen’s book technically defines weak equivalences of simplicial sets to be the morphisms that factor as a trivial cofibration followed by a trivial fibration; only later in Proposition 4 does he prove that this is equivalent to inducing a homotopy equivalence on geometric realizations.
Indeed, my account of Quillen’s book was obviously incorrect. I edited the article on weak homotopy equivalences to add a few more definitions.
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