Dear Mike, I have no objection to anything you actually say. Yes, of course any functor is an anafunctor and, under AC, every anafunctor is isomorphic to a functor. But that is what I meant. It was just that, emphasizing the fact that anafunctors generalize functors (without AC), I wanted to note that distributors (not necesarily representable) also do, even with AC, and that in this alone lies their importance, plus the fact that they can be composed etc. I do not intend to modify your excellent exposition of anafunctors, a subject that I "learnt" just by reading it. I did not mean it as a criticism. Many thanks, Marta
Date: Sat, 29 Jan 2011 11:02:19 -0800 Subject: Re: categories: Re: Fibrations in a 2-Category From: mshulman@ucsd.edu To: marta.bunge@mcgill.ca CC: categories@mta.ca
Dear Marta,
The discussion of the equivalence in the nLab article you mention was added 5 days ago by me, by extracting and condensing a bit from Jean's, my, and David's emails a week ago. I thought this discussion interesting enough that it ought to be preserved.
On Sat, Jan 29, 2011 at 9:45 AM, Marta Bunge <marta.bunge@mcgill.ca> wrote:
In that article, it is furthermore pointed out that each version has its advantages over the other, and that therefore both are of interest for category theory in a topos S without AC, where they generalize ordinary functors. But, even in the presence of AC, distributors (profunctors) generalize ordinary functors, a fact that I have known for 45 years, whereas anafunctors do not.
This confused me for a minute until I realized you probably meant "generalize" to mean "strictly generalize." (It is of course still true under AC that every functor is an anafunctor; the difference being rather that under AC every anafunctor is isomorphic to a functor.)
As David pointed out, since the nLab is a wiki, anyone who feels the discussion currently existing there has defects should feel free to take it upon themselves to remedy those defects.
Best, Mike
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