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I look at ind-cocompletion and pro-completion issues these days. New entry cocompletion. References at many related entries. Notably (at inaccessible cardinal)
We consider four notions of strong inaccessibility that are equivalent in ZFC and show that they are not equivalent in ZF.
(Strange: if I paste ZF and ZFC in font from the Fundamenta page abstract, the nForum truncates everything starting with ZF. This way I lost part of the text which I wrote after this.)
Note that the wikipedia and some other sources have an outdated link for the Blass et al. paper, at a Dutch site. This pdf link is to the Polish Fundamenta site, and works as of now.
In my opinion, pro-representable functor and pro-object should be just chapters in the same entry. But let us leave that for now.
They do capture different aspects of the same idea, and each can be seen as an entry point for studying the other as well. It should be made clear in the entry that this is the case, but merging the two would to some extent muddy the waters on this.
I think it is the opposite: much of unnecessary work and multiple confusions in the literature (up to present day) are thanks to separating the notions (and schools) which are equivalent by directly expressable equivalences.
I have some sympathy with the approach you put forward, and way back I argued in Shape Theory that taking the comma category of each space with respect to the polyhedra (i.e. , for ) was a good way to look at what shape theory was telling us, BUT there were highly respected voices amongst the shape theorists who strongly opposed that view saying that it was confusing. They thought of spaces and inverse systems as being geometric and could not bridge the tiny gap to the more categorical veiewpoint. They wanted a highly concrete presentation of the theory, not an abstract one, even though, IMHO, the abstract one said what was going on much more clearly.
My point about muddying the waters is that researchers approach the area from different starting points and with different backgrounds. For some using prorepresentable functors would be a complete turn off whilst inverse systems would seem more friendly. My view on the entries is that we should aim to make it clear that both viewpoints can be useful, but that the equivalence between them is the key to understanding.
On a related theme, in proving results on profinite groups sometimes I find the categorical apprach is by far the clearest, but then for other results, a topological approach allows faster progress. They are equivalent but this is playing with the equivalence as a way to clear away confusion.
FWIW, I think it is very much in the nPOV to have only one entry about things which were historically kept separate for sociological reasons but which nowadays we can see as the same.
Perhaps we need a main page on pro-objects as functors, then two linked pages discussing the two approaches, emphasising the equivalence. (Having too long a page does not always help the reader, and merging pages has sometimes resulted in confusing points of view on the single entry.)
I would really love someone to write out a description of the Grossman-Isaksen model category structure on pro-categories, using just the pro-representable functors appraoch, or to clear up Quick’s model category structure on simplicial profinite spaces (which is a bit of a mess in places). In both cases it probably is something very neat and a restriction of some general case when viewed in the right way.
Tim, you know that moreover at the “pro-object” side there are at least 4 different approaches (two with quantifiers in morphism definition and no need for equivalence classes and two with using choice to make index function but then with need for equivalence classes; two are with inverse systems two with filtered systems)…
Besides there is also pro-completion given by universal property, without direct construction by contravariant Yoneda or by limits of filtered functors…
Zoran: you are, of course, right. I have lived and worked with and in that mess for most of my research life. My problem is that there are still people who will use the less robust theories and will swear that that is the ’correct’ thing to do. In order to ’talk. to them, I have used ’pro-objects as formal limits’ as a model for the concept rather than pro-representable functor. (I think my views on this have evolved as well during the evolution of the concepts.) To hope for some sort of impact I tried to make my papers readable by the people working in the area, but that means using their language! Taking that forward to the nLab gets us to the question of the approach that should be put across (initially with as little change to the existing material as possible, as I have not got that much time as my editing for Ronnie Brown’s birthday issue of JHRS, plus several talks to prepare, etc. seems to take up a lot of time.) Another problem is that as, for instance, the material on prohomotopy (even Edwards and Hastings stuff) was written within the pro-object style, it will require some thought as to what that looks like via pro-represntability. The pay-off for such a pice of work is also not that clear although it might clarify the theory no-end.
There are also the links with Proper Homotopy (an area which might benifit form an n-Lab treatment using pro-representability) but which fits well within the pro-object setting at the moment.
There was a shape theorist (now dead) who did not like that pro-object approach and argued with me that the approach was unintuitive. My viewpoint at the time was that the pro-objects were like the Cauchy sequence approach to rational numbers, and therefore were a useful way of looking a the situation, but although competant with category theory, that shape theorist could not make feel happy with that analogy.
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