In May and Sigurdsson "Parametrized homotopy theory" there is a general treatment of Hurewicz style model structures in Chapter 4, see definitions 4.2.1 and 4.2.2. I am trying to adapt these to a more general setting. There is an "observation" after Lemma 4.2.4 stating that $X \to \mathrm{Cyl}(X)$ and $\mathrm{Cocyl}(X) \to X$ are strong Hurewicz cofibration and fibration, respectively. The text makes it sound as if this holds for a purely formal reason, i.e., no geometry of the interval is used. Is this really the case? I do not see this, and it is crucial for what I am trying to do to know the exact geometric requirement on the interval for this to hold. I think I am missing some fairly obvious use of the lifting properties, and my excuse is that I am not a homotopy theorist.
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1$\begingroup$ I'm too sleepy to be sure, but this paper might help. $\endgroup$– Omar Antolín-CamarenaCommented Jul 6, 2013 at 8:00
2 Answers
Andrej, thank you for your reading of May and Sigurdsson, which I wish we had made more accessible. I don't have time to check anything, but my recollection is that the verification of the observation is easy, whether or not it is formal, and works equally well in homological analogues. I think we inserted it only for reassurance, since it is of course to be expected. It is obviously true when spaces are restricted to be compactly generated since then there is no ``strong'' distinction. (MS could not just restrict like that because of point-set problems with parametrized function spaces).
The paper of Schw\"anzl and Vogt noted in Karol's answer is the original source for the distinction between strong and ordinary Hurewicz cofibrations and fibrations and may well be helpful. A crucial non-formal thing, noted just after the observation you refer to and emphasized more strongly in Schw\"anzl and Vogt is the need for a good cylinder object: $\partial I \longrightarrow I$ has the LLP wrt $h$-acyclic $h$-fibrations. (Of course, that fails for simplicial sets.)
In case you are thinking about model structures, I'll say a bit about that (probably repeated from another comment or answer). There is a mistake in the general proof of the factorization axioms in the paper of Cole that Karol refers to and in MS (4.4.2) and MP ("More Concise''). A fix is given in: T. Barthel and E. Riehl. On the construction of functorial factorizations for model categories. Algebraic & Geometric Topology 13 (2013), 1089--1124. (That is the paper referred to by Omar, but their fix does not seem directly relevant to the precise question you ask). They and I are just finishing a related paper "Six model structures for DG-modules over DGAs'', in which their fix is again utilized and is described in full categorical generality. If anyone is interested, I will be happy to say why ``six'' and why that is also to be expected in topological contexts of interest (but I sometimes get scolded for digressing).
Ok, Karol, but briefly: Six projective type model structures should appear whenever has a category $\mathcal M$ of structured objects enriched in a category $\mathcal V$ with two model structures (like the $h$- and $q$-model structures on spaces, $R$-modules, and, conjecturally, certain categories of spectra). The category $\mathcal M$ then has three natural subcategories of weak equivalences, the structure preserving homotopy equivalences ($h$-equivalences), the homotopy equivalences of underlying objects in $\mathcal V$ ($r$-equivalences, where $r$ stands for relative), and the weak equivalences of underlying objects in $\mathcal V$ ($q$-equivalences). These can be expected to yield $q$-, $r$-, and $h$-model structures with accompanying mixed $(r,h)$-, $(q,h)$-, and $(q,r)$-model structures. And they are interesting!
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$\begingroup$ Thanks for the comments. I was aware of the problems you mention in the last paragraph, as my starting point was Barthel & Rhiel. I might as well explain what I am doing. I am trying to get Hurewicz style model structure inside extensional type theory, or in categorical terms in a locally cartesian closed category. This in particular means no excluded middle in general. The consequence is bizarre: paths cannot be concatenated as usual. But I have a cunning plan (which involves the lccc structure in an essential way), which however involves checking a lot of details. $\endgroup$ Commented Jul 7, 2013 at 6:29
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1$\begingroup$ I, for one, would like to read more about these six model structures. $\endgroup$ Commented Jul 7, 2013 at 8:35
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$\begingroup$ Just to make sure: by underlying (weak) homotopy equivalences in a $\mathcal{V}$-enriched categories do you mean morphisms inducing (weak) homotopy equivalences on all (fibrant?) representables? Or does it depend on the notion of a "structured" object? $\endgroup$ Commented Jul 8, 2013 at 5:21
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$\begingroup$ I assume that I have a forgetful functor, U say, from M to V. A map f in M is an r-equivalence if Uf is an h-equivalence. It is a q-equivalence if Uf is a q-equivalence. In our algebraic context, V has h and r model structures that happen to coincide, which makes the definition look more natural. $\endgroup$ Commented Jul 8, 2013 at 17:03
An explicit proof can be found in Cole's Many Homotopy Categories Are Homotopy Categories (Lemma 3.4). Some topological properties of the interval are used and it seems crucial to the argument. These properties are essentially that $I \times I$ retracts onto both $I \times \{ 0 \}$ and $I \times \{ 0 \} \cup \{ 0, 1 \} \times I$ and that $(I \times I, I \times \{ 0 \})$ and $(I \times I, I \times \{ 0 \} \cup \{ 0, 1 \} \times I)$ are homeomorphic as pairs of spaces.
This doesn't hold e.g. for the simplicial interval and it seems that constructing Hurewicz type model structures on simplicial categories is much more subtle. Some theory for pretty general enrichments is developed in Strong Cofibrations and Fibrations in Enriched Categories by Schwänzl and Vogt.
EDIT: I have realized that I misread your question. You were asking about the projection $X^I \to X$ while the above applies to $X^I \to X \times X$ which is more subtle. With Cole's definition of strong fibrations it is obvious that $X^I \to X$ is one. May and Sigurdsson use a slightly different definition but Cole proves that they are actually equivalent (see my comments below).
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$\begingroup$ I am sorry but this does not seem very helpful. In my case the starting point is a general cofibration $i: A \to X$, whereas Lemma 3.4 in Cole considers a specific case which is easily related to the geometry of the interval. $\endgroup$ Commented Jul 6, 2013 at 13:32
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$\begingroup$ I don't understand your remark. In the proof of Lemma 3.4 Cole starts with the case of an arbitrary acyclic cofibration ($g : Z \to W$ in his notation) and then reduces it to "a specific case which is easily related to the geometry of the interval". The reduction itself also heavily uses topology of the interval. $\endgroup$ Commented Jul 6, 2013 at 15:09
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$\begingroup$ Maybe this will clarify things. Actually, it's Cole's definition that is a priori stronger than May's and Sigurdsson's. For Cole a strong fibration is a map with RLP with respect to all acyclic cofibrations, for May and Sigurdsson it is one with RLP with respect to pushout products of arbitrary cofibrations with $\{ 0 \} \to I$. Cole proves that every such pushout product is an acyclic cofibration (Prop. 2.10) and that every acyclic cofibration is a retract of such a pushout product (Prop 2.13)... $\endgroup$ Commented Jul 6, 2013 at 15:53
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$\begingroup$ ... Thus the definitions coincide and Lemma 4.3 indeed proves that $X^I \to X \times X$ is a strong fibration in both senses. $\endgroup$ Commented Jul 6, 2013 at 15:54
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$\begingroup$ Ok, thanks, I will have another look. Your comments are helpful. $\endgroup$ Commented Jul 6, 2013 at 18:10