Hi Jean, I reply only to your point 3, which I reproduce:
3- You also say:
"An anafunctor is really a simple thing: a morphism in the bicategory of fractions obtained from Cat by inverting the functors which are fully faithful and essentially surjective".
Woaoo, you call this a simple thing! Ordinary categories of fractions are very complicated, unless you have a calculus of right (or left) fractions. Is there, precisely defined, and without neglecting the coherence of canonical isomorphisms, such a "calculus" defined. Does it apply to the "simple thing" of anafunctors.
Yes. Dorette Pronk, in her 1996 JPAA paper, defines conditions on a bicategory B equipped with a class of 1-arrows W such that the localisation B[W^-1] is defined and is given by a calculus of fractions. In chapter 1 of my PhD thesis (reproduced in http://front.math.ucdavis.edu/1101.2363) I prove the following: Let S be a category with binary products and equipped with a singleton Grothendieck pretopology E satisfying - if A -> B is a split epi in S and A-> B -> C is in E, then B-> C is in E. Then consider a full sub-2-category Cat'(S) --> Cat(S) that is closed under the operation X |-> X^I where the latter is the internal category of isomorphisms, pullbacks of source and target maps of all categories which are objects in Cat'(S) exist, and there are cartesian lifts of arrows in E along the underlying objects functor _Cat'(S)_ --> S (_Cat'(S)_ is the underlying 1-category). Then: Cat'(S) admits a calculus of fractions for the class W_E of 1-arrows which are fully faithful and essentially E-surjective (meaning, the usual map in S expressing ess. surj. is in E, surjectivity meaning nothing in arbitrary S), and furthermore, these are calculated by anafunctors. There are many examples in the literature (even Pronk's own paper) where various classes of internal groupoids are considered and localised at W_E for various S and E. Algebraic, topological, Lie groupoids, and 2-groups (=groupoids internal to Grp) are considered among others (locally compact topological groupoids equipped with Haar measures, general abelian or semiabelian categories in place of S and so on). I believe all of the results about localisation in these settings are captured by the above result, though I have not checked them all (or even found all such results). Many of them assume finite completeness of S or similar, but it is not necessary (indeed when dealing with S=Manifolds, this is obviously not present). To forestall one criticism, there are certain anafunctors A<--C-->B internal to S (the saturated ones) which can be written down as an internal discrete opfibration gC --> gA x gB, (gA is the underlying groupoid of the category A) which if I guess correctly are one correct notion of internal distributor. However, one cannot compose these in all cases, as one can anafunctors - some well-behaved quotients are necessary, as I'm sure you would agree. I'm considering more general categories, and I do not know (and I should try to find out!) if general anafunctors correspond to internal fibrations. I would be a happy occurrence if they did. (On a personal point, I'm only interesting in internal categories. My 'usual category theory' I like with Choice, and liberal use of it sometimes, so I'm not advocating, necessarily, new AC-free category theory, although my results may be helpful there.) I'm not dismissing the usefulness of distributors, but I find that for internal purposes, anafunctors are a helpful tool and provide good perspective. I only provide one example, because my breakfast is getting cold: anafunctors compute 1-dimensional 'nonabelian cohomology' for practically any case you care about (if not all cases). Higher-dimensional generalisations of anafunctors compute all the usual forms of cohomology (or practically all, not every single case of cohomology in the literature has been checked to be given so), but this is not what I want to cover, so I leave it there. It is precisely the presence of the calculus of fractions that anafunctors afford (and they simplify Pronk's construction) that I believe reflect their utility. Best regards, David [For admin and other information see: http://www.mta.ca/~cat-dist/ ]