22 Jan 11:25 2011

## Re: Fibrations in a 2-Category

JeanBenabou <jean.benabou <at> wanadoo.fr>

2011-01-22 10:25:55 GMT

2011-01-22 10:25:55 GMT

ANAFUNCTORS VERSUS DISTRIBUTORS Dear Mike, (I apologize for using in a few places capital letters, where normally I would have used italics, but html is not accepted in the Category List) In your mail about fibrations in a 2-category, dated Jan.14, you say: "One way to deal with the difficulty you mention is by using "anafunctors," which were introduced by Makkai precisely in order to avoid the use of AC in category theory". There is "another way", which I prefer. It is using distributors, which do much more than merely "avoid the use of AC", and apply to more general situations than the ones you consider. Let me first give a very simple definition: Let M: A -/-> B be a distributor, identified with a functor A --> (B °, Set) = B^. I say that M is "representable" iff for every object a of A the presheaf M(a) is. With AC, such an M is isomorphic to a functor F: A --> B, which is unique up to a unique isomorphism. But my definition doesn't need any reference to AC. I shall denote by Rep(A,B)) the full subcategory of Dist(A,B) having as objects the representable distributors. "Corepresentable" distributors are defined by the canonical duality of Dist, and I denote by Corep(A,B) the corresponding category. 1- In your example you say: "let P --> 2 be a fibration, with fibers B and A. Then there is (without AC) an anafunctor A --> B, where the objects of F are the cartesian arrows of P over the nonidentity arrow of 2, and the projections assign to such an arrow its domain and codomain" What I can say with distibutors is: 1' - Let P --> 2 be an ARBITRARY functor with fibers B and A. Then there is, without AC, a canonical distributor A -/-> B associated to this functor. Moreover the the functor is a fibration iff the associated distributor is representable, and a cofibration (I think you'd say "op-fibation") iff this distributor is corepresentable. . (again no AC). Which statement do you prefer? 2- A little bit further on you say: "More generally, if Cat_ana denotes the bicategory of categories and anafunctors, then from any fibration P --> C we can construct (without AC) a pseudofunctor C^{op} --> Cat_ana." With distibutors I can say: 2' - Let F: P --> C be an ARBITRARY functor. From F, I can construct, without AC, a normalized lax functor D(F) : C^(op) --> Dist . Then we have, without AC: (i) F is a Giraud functor (GIF) iff D(F) is a pseudo functor. (ii) F is a prefibration iff for every map c of C the distributor D(F) (c) is representable (iii) F is a fibration iff it satisfies (i) and (ii) (Iv) F is a cofibration if it is a GIF and the D(F!(c)'s are corepresentable. Which statement do you prefer ? In (iv) I insist on the fact that it is the same D(F). Is there a notion of "ana-cofunctor"? Note moreover that many other important properties of F can be characterized by very simple properties of D(F), again without AC! 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. 4- In guise of conclusion you say: In general, it seems to me that there are two overall approaches to doing category theory without AC (including with internal categories in a topos): 1) Embrace anafunctors as "the right kind of morphism between categories" in the absence of AC 2) Insist on using only ordinary functors, so that we can work with the strict 2-category Cat, which is simpler and stricter than Cat_ana. "Personally, while there is nothing intrinsically wrong with (2), I think (1) gives a more satisfactory theory." Sorry,but your approaches 1) and 2) are not the only ones. I opt for the following one: 3) Work with distributors. I still have to see precise mathematical applications anafunctors.. Do I have to mention applications of distributors? Do I have to point out that distributors can, not only be internalized, but also be "enriched"? 5 - You are a very persuasive person Mike, but I'm not "buying" anafunctors, unless you give me convincing examples of what anafunctors can do, which distributors cannot do much better. And if you want to generalize functors, without going all the way to arbitrary distributors, good candidates, for me, instead of anafunctors, are representable distibutors, which are very simple to define rigorously and easy to work with. And of course don't use AC., I have a very strong guess that anafunctors are "the same thing" as representable distributors. I can even sketch a proof of my guess. (i) You say that an anafunctor can be represented by a span A <-- F -- > B where the left leg, say p, is full and faithful and surjective on objects and the right leg, say q, is arbitrary functor. In Dist you can take the composite: q p*: A -/-> F --> B, where p* is the distributor right adjoint to the functor p. It is easy to see that his composite is representable. Thus we get a map on objects, u: Cat_ana(A,B) --> Rep(A,B) (ii) Conversely, suppose M: A -/-> B is representable. By 1' we get a fibration P --> 2 thus by 1 an anafunctor A --> B . Thus we get a map on objects, v: Rep(A,B) --> Cat_ana(A,B) . It should be routine that u and v extend to functors U and V and give an equivalence of categories between Rep(A,B) and Cat_ana(A,B) I didn't write a complete proof because, in order to do so, I'd have to know a little more than what you wrote about the category Cat_ana (A,B) and I'm not ready to spend much time on the study of anafunctors. Is my guess correct? If it isn't, where does my "sketch of proof" break down? In particular what is the category of anafunctors with domain the terminal category 1 and codomain a category C? I'd be very grateful if you could answer these questions, and some of the ones I asked in 1) and 2). I'm sure I didn't convince you. All I hope for, is that a few persons, after reading this mail, and your future answer of course, will think twice before they abandon "old fashioned" Category Theory with its functors, AND DISTRIBUTORS, and rush to anafunctors, with the belief that they are unavoidable foundations for the future AC- free "New Category Theory". Regards, Jean, [For admin and other information see: http://www.mta.ca/~cat-dist/ ]