Re: Fibrations in a 2-Category

[Marta Bunge <marta.bunge@mcgill.ca>]

Dear Jean (and Mike),
Your guess/proof that anafunctors and representable distributors are equivalent notions, and by an equivalence which extends to an equivalence of categories between Rep(A,B) and Cat_ana(A,B), is not only correct but  also fully discussed in:

http://ncatlab.org/nlab/show/anafunctor

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 ought to be pointed out by the authors of the article mentioned above. 
In my thesis ("Categories of Set-Valued Functors", University of Pennsylvania, 1966), inspired by a monograph of Michel Andre ("Categories of Functors and Adjoint Functors", Batelle Report, Geneve, 1964), I discuss what you later called "distributors", including their composition as a generalized matrix product. It was Bill Lawvere who pointed out to me the  importance of  "distributors", well before you introduced them. I gave  an expose of a portion of my thesis at the Oberwolfach meeting in 1966. My  main interest therein was the equivalence between the category of profunctors between two small categories and that of those functors between the corresponding set-valued functor categories that have an adjoint or a coadjoint. In turn, this led to Morita equivalence theorems. The results are valid for an arbitrary (co)complete topos S but, even in the case of Set,
which satisfies AC, this shows that profunctors generalize ordinary functors. 

With best regards to all,
Marta

  

> To: mshulman@ucsd.edu; categories@mta.ca
> From: jean.benabou@wanadoo.fr
> Subject: categories: Re: Fibrations in a 2-Category
> Date: Sat, 22 Jan 2011 11:25:55 +0100
> 
> 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,
> 
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