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I have taken this opportunity to update the references section at profunctor, based on recent emails from Marta Bunge and Jean Benabou.
I have added a little detail to the comment at anafunctor that Kelly considered anafunctors without naming them, namely the paper and the year, and also a small concession to Jean Benabou who wanted it widely known that he recently discovered the equivalence between anafunctors and representable profunctors viz, naming him explicitly at the appropriate point of the discussion.
(I do not want to drag the recent discussion held on and off the categories mailing list here - I just wanted to make the changes public)
Thanks, David. Some points
I have made in the $n$Lab page the reference to Kelly’s article an item in the list of references and linked to it from the main text.
What exactly does Kelly consider? As I once tried to emphasize in the section “Homotopy theoretic interpretation”, the idea of regarding spans whose left leg is a weak equivalence as generalized morphisms is very old and very common. And commonly unnamed. I doubt that Kelly is the original source for this (if that’s what the article considers, I haven’t seen it). Maybe we should try to really find some very original sources for this.
Where it says in the $n$Lab entry that $Prof_{rep}$ and $Cat_{ana}$ are equivalent, it says that $Prof_{rep}$ is the “locally full sub-2-category” of $Prof$ on the representable profunctors. Shouldn’t it say the “full sub-2-category” (meaning that the inclusion is an equivalence on each hom-category)? Or otherwise one would need to name on which objects and which morphisms the inclusion is supported.
There has been a leftover discussion that had been sitting in the $n$Lab entry. I think it would be good to resolve in discussion here whatever still needs to be resolved and then incorporate back whatever stable insight has been gained in the discussion as plain text into the entry (not in a query box). That makesthe entry more useful to read. Here it is:
+– {: .query} David Roberts says: If one uses a coverage, then composing anafunctors means a choice has to be made in the filler of $U \to D_0 \leftarrow V$ with the right map a cover. Presumably the resulting bicategory of anafunctors is independent, up to biequivalence, of the choices made. Also, at the very least the identity map has to be a cover, so as to define the identity anafunctor.
DR says: Well I suppose we could follow Makkai’s philosophy twice and have a composition anafunctor (in the original sense) for composing anafunctors (in the internal sense) and end up with an anabicategory.
Mike: Yes, presumably it won’t depend on the fillers chosen; I haven’t checked the details, though. “Grothendieck coverage” means the same as “Grothendieck topology” and thus includes closure under lots of things, including composition and containing identities.
David R: I noticed the adjective Grothendieck in the preceeding sentence half-way through asking the question, but I think my point still holds for general coverages, without the closure properties.
Mike: Well, as you pointed out, you need at least identity maps to be covers to have identity anafunctors, and you need covers to be stable under composition, as well as pullback, in order to define composition of anafunctors. An arbitrary coverage might not satisfy those, although pretty much any coverage arising in practice does. The other point that a choice has to be made unless you have honest pullback-stability is certainly true. So probably the most natural-feeling context in which to work is a (possibly singleton) Grothendieck pretopology. I would be happy for this page to ignore the case where covers don’t have pullbacks; I did some rewriting above to reflect this discussion. I think this discussion could now be deleted; feel free to do so if you agree.
David R: After some thought, one could do without the identity anafunctor, and be satisfied with a anafunctor (of the external variety) giving the identity: $1 \to Ana(X,X)$. I think I should move this discussion to a section of its own, and develop these ideas there. As an aside, I think I saw an example of a non-Grothendieck coverage in John and Alex’s smooth spaces paper.
Toby: You only get an anabicategory anyway, because of the choice of pullbacks (unless the structure of the coverage fixes these, as can be done in $Set$).
Anafunctors really should make sense in any site whatsoever (as long as we can compose ananatural transformations, which I guess we can if the site is subcanonical). The trick of getting away with single maps (as one can do, for example, in a superextensive site) is not really necessary. In fact, using a coverage makes the definitions, while more complicated, really look more natural in topological categories. =–
At profunctor I have
added hyperlinks to people’s names;
restructured the subsections of the References-section:
Named “Original sources” the bit with the old articles and the fight about precedence
At “More modern accounts” I moved the reference to Benabou’s 2000 lecture up to the top, and made the pointer to Joyal’s CatLab a fully-fledged reference that states Joyal clearly as an author
Moved the pointers to blog comments by John Baez to a subsection “Expositions”
Finally I have added a pointer to the relation to anafunctors also to the entry profunctor.
it says that $Prof_{rep}$ is the “locally full sub-2-category” of $Prof$ on the representable profunctors. Shouldn’t it say the “full sub-2-category” (meaning that the inclusion is an equivalence on each hom-category)?
A full sub-2-category is determined by a collection of objects. In this case, the objects of Prof and $Prof_{rep}$ and $Cat_{ana}$ are all the same, namely categories. A locally full sub-2-category is determined by a collection of morphisms, i.e. a full subcategory of each hom-category. The objects of the hom-categories in Prof are profunctors, and we take the full subcategory determined by those that are representable.
Kelly considers exactly what Makkai considers: the problem of defining a functor when one has to choose a ’specification’ for each object. In a sense it is a logical pov. As for the homotopical pov, I think any text in homological algebra that considers resolutions $\widehat{A}_\bullet \to A$ should be considered a candidate. Godement maybe? Leray? Or even Cech-type resolutions…
It really should say locally full sub-2-category, as it contains the same objects, only some of the 1-arrows between these, and all of the 2-arrows between these.
Sorry, right, I made level slip.
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