Re: Fibrations in a 2-Category

[JeanBenabou <jean.benabou@wanadoo.fr>]

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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