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I have split off classical model structure on topological spaces from the entry on “model structures on topological spaces”.
My aim is to have in this entry a detailed, self-contained and polished account of the definition of the standard or classical model structure, its verification and its key consequences.
I have added a fair bit of material today. Not done yet, but I have to call it quits now.
I don’t have any better ideas, but the name we use for this model structure is a bit unfortunate, since some people say “Quillen model category” in general to mean just “model category’.
How about “classical model structure on Top”?
Some people refer to the standard model structure on sSet as the Kan—Quillen model structrure.
Perhaps we could do the same to Top, using Quillen—Serre model structure?
Thanks, that’s a good point. I have added comments and redirects accordingly.
Also I am similarly splitting off an entry classical model structure on simplicial sets from the entry “model structure on simplicial sets”, and will try to add some more love to it. But that should go in its own thread here. Just a moment…
In my mind “classical” would refer to classical homotopy theory, so I would end up thinking about the Hurewicz/Strøm model structure…
I won’t pretend to know much about history of algebraic topology, but I thought that the prevalence of strong homotopy equivalences actually predated the emergence of homotopy theory as a discipline of its own. And by the time this happened, weak homotopy equivalences were more or less accepted as the primary notion. Of course, it’s difficult to make definite statements since some people were ahead of others at various points in time. However, consider that Strøm established his model structure 5 years after Quillen and it was a bit of an afterthought. Strøm himself said that his motivations were “aesthetical” and his result doesn’t seem to have been used a lot by comparison to Quillen’s. Thus I think it’s perfectly fair to reserve the name “classical” for Quillen’s structure.
Thanks Zhen Lin and Karol, for these comments. Please feel invited to add some of this historical discussion into the entry, that would be useful for readers.
For the time being I have changed the entry names to “classical …”. I am not dogmatic about this, but since it takes some tedious work to adjust all the cross-links, I’ll leave it at that for just now.
I find it strange to hear from Karol that homotopy theory was not a discipline before Quillen. Well, it was not abstract homotopy theory, that is his achievement, but the homotopy theory had so many results in 1950s-late 1960-s, and specialists in that subject…
We are not looking for a term for the homotopy theory of the 1950s, but for a term in Quillen’s abstract homotopy theory. His model structure on topological spaces is the first example of a model category that Quillen discusses in his original article, and so it is clearly the classical model category structure on topological spaces in at least one pretty undebatable sense.
I am not dogmatic about this, but let’s try to avoid a heavy bike-shed discussion here. I’d rather you’d all spend your energy on suggesting and making improvements to the content of the entry!
I certainly didn’t mean to say that there was no homotopy theory before Quillen or that Quillen was the first to promote weak homotopy equivalences above the strong ones. All I meant was that by the time homotopy theory was consolidated into something clearly distinct from the rest of topology the focus was already on weak homotopy equivalences. What happened before I would consider “prehistory of homotopy theory” I guess.
In any case, I agree with Urs. Since we are talking about model structures we should consider what is classical from the perspective of abstract homotopy theory not necessarily homotopy theory at large. I guess my previous post would have been less controversial if I had referred to abstract homotopy theory.
The name would not stick, but the homotopy theory post Whitehead should perhaps be ’weak homotopy theory’ as it studies only weak homotopy equivalences. I find the idea that pre-Quillen the homotopy theory was ill-formed somehow a bit sad. The correct name for the theory that comes from Whitehead’s work is probably ’Combinatorial Homotopy Theory’ or ’algebraic homotopy theory’, and the motivation was well explained by JHCW in his ICM talk from 1950.
The term Urs suggests seems great to me. It combines ’classical’ with ’model’ and that feels right to me.
We are not looking for a term for the homotopy theory of the 1950s, but for a term in Quillen’s abstract homotopy theory.
Anyway, it was misleading/confusing to me (and when I forget this in a year or so it will be again…) as it is more or less by definition that Strom’s structure is the model category for classical homotopy theory of topological spaces (that is for the one from the 40-s and 50-s and still existing in basic topology books and based on Hurewicz’s classical definitions of fibrations and cofibrations); the timing of formulation, and minor Strom’s authority in the world of abstract homotopy theory (in comparison to rich Quillen’s work), does not change this essential and intuitive timeless content/meaning/motivation for it. The notions (fibrations, cofibrations and homotopy equivalences) are in Strom’s work not postulated for the first time; he just proved that for the classical (pre-Quillen) definitions of the words the Quillen’s axioms hold (with slight detail: nonpathological closed cofibrations taken instead of all cofibrations). This is I guess accepted when calling it Hurewicz-Strom model structure. It is likely that Quillen knew the fact in the first place, but as the area (properties of Hurewicz fibrations and cofibrations) was already mostly explored he did not bother to write down or even check all the details, hence it did not go into the paper.
Strom’s structure is the model category for classical homotopy theory of topological spaces
And that’s the point we already replied to: in your example “classical” is an adjective for “homotopy theory” and not for “model category”.
I have typed into the entry now the proof of the lemma (here) which asserts that compact subsets are small objects relative to topological cell complexes.
I have taken this straight from Hirschhorn’s (arXiv:1508.01942), just trying to bring out the structure of the argument more transparently.
Just a quick note: I don’t think that #13 is accurate. Strøm’s work is quite non-trivial in several places, and in my opinion the subtleties of it are under-appreciated. One indication of this is the fact that at some point fairly recently, some doubt seemed to arise as to whether Strøm’s work really holds for all topological spaces. This doubt is groundless, Strøm’s work is perfectly valid for unqualified topological spaces.
In particular, I don’t feel that there is any evidence that Quillen ’knew the fact in the first place’. It seems quite clear that Quillen’s axiomatics are an abstraction of the examples given in his original work: chain complexes (leading to the derived category); simplicial sets/abelian groups; and the Serre model structure on topological spaces. The first of these is the most important (with regard to the motivating intuition).
It would be very natural for Quillen to have wondered about the case of Hurewicz cofibrations and fibrations. But wondering about it and having a precise conjecture/some evidence for that conjecture are very different things. It is not the case that one can ’just’ sit down and prove it, one needs quite deep insights into the nature of Hurewicz cofibrations.
I once heard from someone who was around in Norwegian mathematical circles in the 1960s that Strøm’s work was ’assigned to him’ as an ’exercise’ by his supervisor, Per Holm. But this should be taken with a large pinch of salt, I think that the creativity of Strøm’s work may not have been fully understood by the person I heard this from, or by many working in that milieu. It should be said, though, that Strøm’s work on cofibrations would have been more fashionable then that it is now, and would not have ’come out of the blue’.
Having said all that, I don’t think that referring to the Serre model structure on topological spaces as the ’classical’ model structure is a very good idea!
I don’t think it … is a very good idea!
Why?
(compare here :-)
But really, it doesn’t have to be a very good idea. It is sufficient if it is a mildly good idea. The reason is that in any case, it is a tiny idea, an $\epsilon$ of an idea. It’s just a choice of terminology that one has to make, and no choice will make everyone happy. If you all feel strongly about it, I invite you to edit the entry titles or whatever. Just don’t break my links, please, as I don’t have time to fix them all once again.
It is indeed just a minor matter, which is why I did not elaborate. Since you are the person principally making use of the entry and putting the effort into it, I would think you should feel free to make whatever choice you like!
Since you ask, I just feel that ’classical’ is entirely uninformative in itself. I myself refer to the two as the Serre and Hurewicz model structures, because this gives an unambiguous way to tell which is which: the name refers to the type of fibration. I don’t see any need to embelish the two names with ’Quillen’ and ’Strøm’ respectively.
Richard, it’s not that I don’t understand that “classical” has disadvantages. But what I feel is not receiving due attention from commenters here is that any other choice has evident disadvantages, too! This is why I don’t understand how some of you express so strong feelings about what you deem “the right” terminology or notation is.
Looking back through this thread, I feel this is a pure example of the bike-shed effect, and we need to try to steer clear of that more here on the nForum, not to get bogged down.
Back to the entry; meanwhile I think that I am done with spelling out a complete self-contained proof. The last thing I added was the full proof of lemma 5 (which together with lemma 4 are the only two technical lemmas that use the nature of topological spaces, the rest is formal model category theory).
There are still some places that could be streamlined further, and certainly there will be typos and maybe other imperfections left, these I will look into next week.
I have added a section here establishing the projective enriched functors, $[\mathcal{C}, Top_{Quillen}]_{proj}$, as a corollary of the existence of $Top_{Quillen}$.
(The section is mostly just an quick recollection/exposition of topologically enriched functor categories and highlighting the Yoneda lemma, followed by the observation that from this and by just pointing to the previous proof of $Top_{Quillen}$, the result is immediate.)
I have added more details (in Related model structures) on the induced cofibrantly generated model structures on a) pointed topological spaces, b) compactly generated topological spaces and c) topologically enriched functors.
I have added discussion of the topological enrichment of $(Top_{cg})_{Quillen}$ here.
I have renamed the section to Monoidal and topologically enriched model structure and rearranged slightly. Then I have expanded and polished further: said a few more words about the proof that $I_{Top} \Box I_{Top}\subset I_{Top}$ and $I_{Top} \Box J_{Top} \subset J_{Top}$ (copied that over to the Examples at pushout-product), spelled out the proof of the “Joyal-Tierney calculus” more, and added, at the end of the section, quick discussion that all goes through directly analogously in the pointed case.
added publication data to:
Redirect: Serre model structure.
Used here: https://doi.org/10.1016/j.jpaa.2005.12.010 by K. Worytkiewicz, K. Hess, P. E. Parent, A.Tonks.
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