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Zoran just wrote Hurewicz fibration.
I’ve added the abstract definition by Michael Warren. Is this the only abstract treatment?
I’m sure something similar is defined in [Kamps and Porter].
An abstract treatment is certainly in Kamps and Porter, but is, relatively speaking, ancient history (and also absolutely obvious once has an abstract notion of interval)! It is explicitly in Kamps’ work from around 1970 (see 1.7 in Kan-Bedingungen und abstrakte Homotopietheorie), and also essentially present in Quillen’s book in which model categories are introduced, from roughly the same time. It also appears extensively in work of Grandis.
You can also find it in my thesis, where in addition you can find an axiomatisation in the setting of an interval/cylinder and co-cylinder of what I call ’normally cloven fibrations’ (and cofibrations), which are essential to consider when working constructively, and in particular when modelling Martin-Löf-like type theories.
Richard, if you find a second, it would be nice if you could add this kind of remark into the entry Hurewicz fibration. Thanks!
Another good reference is Schwänzl and Vogt, “Strong cofibrations and fibrations in enriched categories”, 2002.
This MO question by Andrej also seems relevant.
In case it is of interest, regarding the Math Overflow question and strong cofibrations/fibrations: the latter are closely related to ’normally cloven cofibrations/fibrations’ in the sense of my thesis, but to see this is non-trivial, and one needs one’s interval to be equipped with certain ’structures’. A much older (again, around the early 1970s) terminology for strong fibrations is: a map satisfying the covering homotopy extension property. I would regard ’normally cloven fibrations/cofibrations’ as the correct things to take as one half of a Hurewicz-type model structure, and would regard the ’covering homotopy extension property’ rather as a (crucial!) tool which one uses along the way to obtaining the model structure.
With regard to obtaining Hurewicz-type model structures constructively, as Andrej was looking to do, my thesis gives the only general way that I know of to do this (the approach is constructively valid in a very strong sense). The use of ’normally cloven fibrations/cofibrations’ is, as I mentioned in my previous comment, fundamental to this approach. The significance of the conditions defining a normally cloven fibration/cofibration was discovered independently by myself (for establishing the lifting axioms for a model structure) and by Garner and van den Berg (in Topological and simplicial models of identity types, for establishing the factorisation axioms for a model structure), and I would be extremely surprised if there is any general, constructively valid method for constructing Hurewicz-style model structures which does not involve consideration of them.
The folk model structure on $\mathsf{Cat}$ can, for instance, be constructed constructively, but one has to work with either normally cloven iso-cofibrations or normally cloven iso-fibrations (so that in fact one has two model structures, slightly different from a constructive point of view). Whenever one is working constructively with model structures, one also has to forget about cofibrant generation, and find other tools instead.
Hi Urs, thank you for the invitation, but editing the nLab is not something that I am planning to do in the near future. I realise that this is likely to be frustrating, my apologies, but somehow making small edits here and there is not something I enjoy/am motivated to do, and something more major is not possible for me, at least for the foreseeable future.
I have edited it in here.
making small edits here and there is not something I enjoy/am motivated to do
You already did precisely that in #4 and #8 above. The energy is more sustainably spent by making the same keystrokes on the $n$Lab. There people will find your pointers, while this thread here will soon be lost.
I have added some links and text covering some of the points made earlier.
Thanks!
In reply to #10: for me, writing something in the nForum feels very different to making a small edit on the nLab. On the nForum, I can just write down what I wish to say without having to think about anything else. For me it is not important whether or not something I write on the nForum will be preserved in the long run, I just see it as a contribution to a discussion, as one might contribute to a discussion between people face-to-face: for most people, a pre-requisite to participating in such a discussion is not that someone has a tape-recorder to preserve what is said, but the discussion might be useful to some of the participants nonetheless!
There is also a different feeling that I just do not think a wiki such as the nLab is for me. To give an example: the one time that I did make a substantial edit on the nLab, to the factorization lemma page, it was mostly written over again within a few weeks. Now that’s absolutely fine, that’s how a wiki such as the nLab works, and what is there now no doubt fits better with the rest of the nLab than what I wrote; and I can find what I wrote by looking through the revision history, so I have not wasted any effort. But somehow, when I write mathematics, I feel that I have a certain ’voice’ that makes it my own, even if that ’voice’ comes from within me and is not distinguishable to anybody else. I expect that many or most mathematicians feel this way: it is what gives one’s mathematics its authenticity. And, for me at least, feeling that the mathematics that I write is authentic to my ’voice’ is important to my enjoyment of doing that writing.
To try to explain a little what I mean by a ’voice’: a serious Leonard Cohen or Bob Dylan fan will almost always recognise any one of their songs as being written by them within a few seconds, even if they cannot pinpoint the exact song without listening to it longer, because they recognise the unique ’voice’ of these musicians: I am not referring to how they sound when they sing, but a certain, impossible to describe exactly, authenticity which one ’feels’ in their work, and characterises it as coming from them. An artist of any kind has to find a ’voice’ of their own, and I feel it is the same for mathematicians.
To get back to the nLab: somehow I just don’t feel that I can express the ’voice’ which makes writing mathematics enjoyable for me in a wiki of the nature of the nLab. This is the case even for small edits. Thus I do not enjoy it, and so I prefer to make a definite decision not to do so at all, and not to worry about it further. And that is not a criticism at all of the nLab, which I find extremely useful: I indeed use it all the time. Writing in it is just not for me. Fortunately, many other people are very good at working on wikis such as the nLab, and very happy doing so, and that is great. It takes all sorts to make a world…!
Hi Richard,
sure, I understand. Nevertheless, when you add valuable information to the nForum you’ll have to live with me asking you to add something to the $n$Lab every now and then. It would be the same if you added good information on Wikipedia’s talk pages, and the $n$Forum is just the talk pages of the nLab. Feel free to ignore me, but I’ll keep prodding you :-)
Regarding what you say about your edits being “overwritten”, that shouldn’t happen, at least not without discussion. Let’s sort this out in the relevant thread.
Hi Urs, no problem! Thank you for your understanding!
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