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Jesse Sigal (Jun 25 2025 at 15:42):Hi all, I've been reading Avoiding the axiom of choice in general category theory by M. Makkai, and therer is a citation for a paper in preparation entitled "Anafunctors and indexed category theory" by M. Makkai and R. Paré.
However, it doesn't appear to ever have been published or made public under that name. After searching, I couldn't find any other work by either Makkai or Paré about said topic. Does anyone have any information about this?
Jesse Sigal (Jun 25 2025 at 15:43):Thanks in advance for any and all help!
Mike Shulman (Jun 25 2025 at 16:29):I think I remember looking for that supposed paper myself in the past, also without success. My guess is that it never appeared.
Jesse Sigal (Jun 25 2025 at 16:37):Ah that's too bad, thanks :) I also sadly couldn't find something which was like it spirit (to my level of understanding).
Mike Shulman (Jun 25 2025 at 16:54):I forgot what Makkai said they would do in that paper.
Nathanael Arkor (Jun 25 2025 at 20:09):I would suggest emailing them and asking (and if you do get a response, please share what they say!).
David Michael Roberts (Jun 25 2025 at 21:53):At the very least, I can say that I learned from Bénabou that anafunctors correspond to 'representable distributors/profunctors', and of course these correspond to indexed categories
Jesse Sigal (Jun 26 2025 at 13:24):@David Michael Roberts would you mind elaborating on that please?
David Michael Roberts (Jun 27 2025 at 00:27):@Jesse Sigal I would like to, when I have time. I started looking at the email correspondence where I learned this (much of it public, on the categories mailing list in 2011) and sadly I would rather extract the mathematics than point you at the emails themselves. It was a slightly unpleasant experience, as a junior category theorist at the time. This will need a time of concentration to summarise in an appropriate way. I should probably write a blog post. However, for the present, see this pre-made section on the nLab! (not written by me): https://ncatlab.org/nlab/show/anafunctor#anafunctors_versus_representable_profunctors
Nathanael Arkor (Jun 27 2025 at 06:34):What do you mean by "representable distributors correspond to indexed categories"?
David Michael Roberts (Jun 27 2025 at 06:47):Oh, I think I was typing on mobile and trying to rush. Of course, that doesn't make a lot of sense. I was thinking of taking the span that is an anafunctor, considering it as a functor , and then making that a profunctor. Please trust what is on the nLab over what I wrote above. As I said, the affair was somewhat distressing at the time, I was being publicly excoriated for being exciting about work that extended what I did in my PhD thesis. The original emails have more detail, but I don't want to share those unfiltered.
David Michael Roberts (Jun 27 2025 at 06:51):The point that I was trying to make at the time, and I hope backed up by my paper The elementary construction of formal anafunctors as well as the precursor papers on localisation and anafunctors, is that anafunctors between internal categories can be done internally with very little structure, in particular without the assumption of pullback stable quotients, which is what is needed for internal (representable) distributors/profunctors between internal categories. The response was, roughly, "why do you think I introduced calibrations??", and these are exposed only in a (pair of?) Comptes Rendus notes that are mostly about something else, and don't mention distributors at all.
Classic "I did this in the 70s but didn't publish it, your work is unoriginal" folklore problem. Except perhaps the "folk" in this case was one person.
Jesse Sigal (Jun 27 2025 at 14:03):Thanks David for the nLab link and the further detail! If you feel like it and find the time, I'd love to read such a summary. Sorry to hear about that negative experience, I can only imagine how discouraging that would feel.