Why is the path fibration a strong Hurewicz fibration? 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.
 A: 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!
A: 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).
