Urs: Hi David!
Good idea to post your notes here. Maybe on top of that it would be helpful to post something to the blog in order to attract more people. Maybe just a comment in the entry “General discussion” would do, pointing out that this part of your thesis is exposed here, or, if you like, I could forward a guest post of yours.
I am pretty busy right at the moment and admit that I have just browsed your piece on anafunctor. What I have seen looks very good. You seem to have looked into this very carefully.
Since I haven’t fully read it, the following question may be obsolete. But it won’t hurt to ask it anyway:
categories internal to some category have in particular an incarnations as category-valued presheaves on . From that perspective one would think of them as certain pre-stacks on . If we restrict ourselves to the groupoidal setup, then this is well studied for the case where categories are generalized to infinity-groupoids.
In these contexts, a topology on induces naturally the structure of a homotopical category on -groupoid valued presheaves (some of which may be thought of as -groupoids internal to ) and to that is naturally associated an (infinity,1)-category.
I am saying this because to some extent internal anafunctors can be seen to be a means to represent the morphisms in this higher category. For me, personally, I felt that this perspective helped me understand a bit better the natural role that anafunctors play.
So my question is: have you thought about anafunctors from this perspective? Of course I know that a few months ago we had an intensive discussion about precisely a question of this kind. So I guess I am asking to which extent you find thoughts like that relevant in the notes you present here.
David Roberts: Hi Urs,
I’ve kinda avoided mentioning (or thinking about) presheaves in this setting, because this is a purely internal version of what is going on. Of course, weak equivalences in my sense will become equivalences of presheaves in some sense. Bunge-Pare did this sort of thing for a regular category in 1979, passing between internal and external equivalences (=equivalences of fibred categories over the site).
I do have some ideas relating to the main theme of my thesis in the direction of higher categories. Namely, higher covering spaces, where I use the (possibly Trimblean) topological fundmental n-groupoid, and hence sheaves of such things on . In that setting, the notion of anafunctor you mention is essential.
But that is for the future. For the time being, I’m glad to figured out the sensitivity of the notion of weak equivalence to the chosen pretopology. For constructions, ordinary open covers are sometimes the best, or for general considerations, something with a lot more covers (i.e. a saturated pretopology, but generating the same weak equivalences, may be of use. I was talking with Danny Stevenson a while back and he told me about weak equivalences of simplicial spaces as the maps which are weak equivalences and fibrations in the simplicial presheaf setting for the local Kan structure. This has always been useful to me.
Mathieu: Note that one direction in Proposition 24 of Pronk’s paper (and so in Proposition 2.5.8 in your Chapter 2) is wrong: , which maps a pointed category to the monoid , is the 2-category of fractions of for the fully faithful functors and these form a right calculus of fractions. But, of course, is not locally fully faithful, so condition EF3 doesn’t hold.
But anyway, you need only the other direction. I needed it for something very close to what you do: for an abelian category 1. the 2-category of fractions for weak equivalences (defined using (regular) epimorphisms) in (internal groupoids, functors and natural transformation) is the groupoid-enriched category of internal groupoids, internal saturated anafunctors (defines as functional everywhere-defined internal distributors) and morphisms of distributors; 1. the 2-category of fractions for weak equivalences in (morphisms in , commutative squares and chain complex homotopies) is the groupoid-enriched category of morphisms in , butterflies and morphisms of butterflies; 1. and so, and are equivalent (and are both “good 2-abelian”, in the sense of my thesis).
I wonder what is the most general context in which one can define saturated anafunctors. I do it in an exact category, using regular epimorphisms, since it’s what I needed (for abelian categories), but for example Hirsum-Skandalis morphisms seems to be defined as some kinds of internal distributors, defined in a non-exact category, so I assume that it is possible to do it using a topology satisfying perhaps additional conditions.
David: Thanks for pointing this out Matthieu, I’ll fix that pronto.
As far as saturated anafunctors go, I think that I have a working definition which I’ll put up here soon (soon may be next week!). It basically just says that the wrong-way leg of the anafunctor is a fibration, and not just a surjective (in the sense of the topology given), fully faithful functor.
One nice thing that it seems to imply is this: consider the differentiable category, with surjective submersions as your covers. Then a pre--bundle on a manifold is an anafunctor . The manifold is the total space of a principal -bundle if and only if this anafunctor is saturated. Then the functor to is just given by the ‘difference map’, sending a pair in the same fibre to the element of sending one to the other.
One can see the relation to Hilsum-Skandalis maps, and to the notion of butterfly, which are built out of extensions, not just surjective maps of groups and pulled back crossed modules.