Re: Fibrations in a 2-Category

Re: Fibrations in a 2-Category

[Michael Shulman]

Dear Jean,

As you surmise, anafunctors are more or less the same as representable
distributors.  I apologize for not pointing that out to begin with,
and I'm glad you brought it up.  Specifically, the bicategory Cat_ana
is equivalent to the bicategory Rep of categories and representable
distributors, in the way that you sketched.  So if by

> 3) Work with distributors.

you mean to (among other things) regard Rep as a replacement for Cat,
then I don't really view that approach as very different from my (1).
In particular, it still resolves the issues you raised about internal
fibrations: ordinary fibrations should equally well give rise to
internal (Street) fibrations, in the representable sense, in Rep.

I have no objection to using the language of representable
distributors instead of anafunctors.  And as you point out,
distributors have many other uses -- you don't need to convince me
that distributors are useful and important!  (I feel obliged to point
out that anafunctors can be enriched just as well as distributors can,
but distributors can certainly be used for many things that
anafunctors cannot.)  My experience in mathematics (which is
admittedly much shorter than yours) is that usually, when a given
concept has two equivalent representations, it is useful to know about
both of them and how to go back and forth between them, since each
will have its own advantages and be preferred by different people.

Thus, my answer to your questions "which do you prefer?" is that I
would prefer to have both at my disposal and be able to use either
one.  Without prejudicing anafunctors over representable distributors,
I could rephrase the point I intended to make in my previous email as:
by replacing Cat with a bicategory whose morphisms are "functors
defined up to isomorphism," we can recover many facts about category
theory which classically require AC.  Whether that bicategory is
Cat_ana or Rep is immaterial to the question of "how to do category
theory without AC."


Having said that, I suppose I should nevertheless also say a little
bit about why I like to have anafunctors at my disposal *in addition
to* distributors.  One reason is that sometimes it requires a little
contortion to put something in the form of a representable
distributor.  For instance, if a category A has binary products, then
there is obviously a product-assigning representable distributor P: A
x A -/-> A, defined by

P(a,(a_1,a_2)) = Hom_A(a,a_1) x Hom_A(a,a_2)

But if A has binary coproducts, then in order to define a
coproduct-assigning representable distributor C: A x A -/-> A, then
(as far as I know) one needs to say something like

C(a,(a_1,a_2)) = the set of triples (a_3,p_1,p_2,f) where p_i: a_i -->
a_3 are the injections into a coproduct and f: a --> a_3, modulo an
equivalence relation (a_3,p_1,p_2,f) ~ (a_3',p_1',p_2',f') if there
exists a (necessarily unique iso)morphism g: a_3 --> a_3' commuting
with all the structure maps.

Of course, C is more easily defined as a corepresentable distributor.
But if you want to define a functor that involves both limits and
colimits, like (a,b,c) |--> a x (b + c), then it is not "naturally"
represented as either a representable or a corepresentable one --
although it always *can* be so represented, essentially by passing
across the equivalence Cat_ana = Rep that you sketched.

By contrast, with anafunctors, all of these functors can be
represented "naturally" in analogous ways.  In the first case, we
consider the span AxA <-- P --> A, where P is the category of binary
product diagrams in A.  In the second case, we consider the span AxA
<-- C --> A, where C is the category of binary coproduct diagrams.
And in the third case, we consider the span AxAxA <-- D --> A, where D
is the category of binary coproduct diagrams together with a product
diagram one of whose factors is the vertex of the coproduct diagram.
Note that in each case, the middle category is the category of models
of a "sketch" in A.

Personally, I find the anafunctor way of representing such "functors"
a bit cleaner, and sometimes easier to work with.  Roughly speaking, I
would say that anafunctors are formulated exactly in order to describe
"functors defined up to isomorphism."  Representable distributors, by
contrast, can be described as "functors valued in representable
presheaves."  "Objects defined up to isomorphism" and "representable
presheaves" are *formally* equivalent (without invoking AC), but not
every "naturally occurring" object-defined-up-to-isomorphism is "given
in nature" by the presheaf it represents.  Some are given by the
copresheaf they corepresent; others aren't given directly in either of
those ways.  (But you can guess from the fact that there are lots of
quotation marks in this paragraph, none of this is particularly
formal.  In particular, I don't present it as an argument intended to
convince you to prefer anafunctors over representable distributors,
but rather as a reason why I or someone else might like to think about
anafunctors *in addition to* representable distributors.  If you like,
you can think of an anafunctor as a particularly convenient
"presentation" of a representable distributor.)


I also find it illuminating that amongst all the "classical" facts
about category theory that naively become false without AC, if you
take the single statement "a fully faithful and essentially surjective
functor is an equivalence" and "force" it to be true in a universal
way, then you end up with a world in which all (or at least most) of
the *other* "classical" facts *also* become true again.  As far as I
know, that fact is easiest to express using anafunctors and calculus
of fractions -- although I would be very interested to see a direct
proof that Rep is the result of formally inverting the weak
equivalences in Cat (i.e. a proof that doesn't essentially go by
proving that Rep is equivalent to Cat_ana).


In regards to your other questions:

> 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.

I wouldn't use the phrase "category of fractions" unless there is a
calculus of fractions; otherwise I would probably say "homotopy
category" or "localization".  Calculi of fractions for bicategories,
and the construction of anafunctors thereby, can be found in:

Pronk, D.  "Etendues and stacks as bicategories of fractions",
Compositio Math. 1996

Roberts, D.  "Internal categories, anafunctors and localisations",
arXiv:1101.2363

> what is the category of anafunctors with domain the terminal category 1 and codomain a category C?

It's sometimes called the category of "ana-objects" or "cliques" in C.
  Its objects are diagrams in C whose domain is a contractible
category.  In the setting of internal categories in a topos, it is a
stackification of C.  (Does that count as a precise mathematical
application?)

Best,
Mike


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Fibrations in a 2-Category

[Marta Bunge]

Dear Mike,

I have no objection to anything you actually say. Yes, of course any functor is an anafunctor and, under AC, every anafunctor is isomorphic to a  functor. But that is what I meant. It was just that, emphasizing the fact that anafunctors generalize functors (without AC), I wanted to note that distributors (not necesarily representable) also do, even with AC, and that in this alone lies their importance, plus the fact that they can  be composed etc. I do not intend to modify your excellent exposition of anafunctors, a subject that I "learnt" just by reading it. I did not mean it as a criticism.

Many thanks,
Marta


> Date: Sat, 29 Jan 2011 11:02:19 -0800
> Subject: Re: categories: Re: Fibrations in a 2-Category
> From: mshulman@ucsd.edu
> To: marta.bunge@mcgill.ca
> CC: categories@mta.ca
> 
> Dear Marta,
> 
> The discussion of the equivalence in the nLab article you mention was
> added 5 days ago by me, by extracting and condensing a bit from
> Jean's, my, and David's emails a week ago.  I thought this discussion
> interesting enough that it ought to be preserved.
> 
> On Sat, Jan 29, 2011 at 9:45 AM, Marta Bunge <marta.bunge@mcgill.ca> wrote:
>> 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 confused me for a minute until I realized you probably meant
> "generalize" to mean "strictly generalize."  (It is of course still
> true under AC that every functor is an anafunctor; the difference
> being rather that under AC every anafunctor is isomorphic to a
> functor.)
> 
> As David pointed out, since the nLab is a wiki, anyone who feels the
> discussion currently existing there has defects should feel free to
> take it upon themselves to remedy those defects.
> 
> Best,
> Mike
  		 	   		  

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Fibrations in a 2-Category

[Marta Bunge]

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,
> 
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
  		 	   		  

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Fibrations in a 2-Category

[Marta Bunge]

Dear all,
>>
The discussion about the equivalence between the bicategories of anafunctors and of representable distributors, due entirely to Benabou as mentioned  below, has an additional interpretation in terms of stack completions which may be of interest to some of you.  In what follows, S denotes an elementary topos in the sense of Lawvere and Tierney. 
>>
The first thing I noticed is that a  "representable distributor" from C to D  (in the sense of Benabou) is none other than a functor from  C to D*, where D* is the (intrinsic) stack completion of D.  This follows from the characterization of (intrinsic) stack completions given in my paper "Stack completions and Morita equivalence for categories in a topos", Cahiers de Top. Geom Diff. XX-4 (1979) 404-436. In plain terms, my characterization says that, for any locally internal fibration A over a topos S,  the stack completion A* of A is obtained as the middle term in the factorization of the Yoneda embedding of A into the category of S-valued presheaves on A through its `weakly-essential image'. In the case of a category D in a topos S, its stack completion is that of the fibration [D] over S,   called the "externalization" of D in my paper with Bob Pare, "Stacks and equivalences of indexed categories", Cahiers de Top. Geo. Diff XX-4 (1979) 373-399. 
> >
In principle then, the theory of anafunctors introduced by Michael Makkai  (and with which I am not really acquainted) could be recast more simply and to advantage in these terms, exploiting for this purpose the universal property of stack completions. For categories in S, this says that an anafunctor from C to D is simply a functor from [C] to [D]*. Just as in the case of anafunctors, for a general topos S,  the fibration [D]* over S for a category D in S need not be equivalent to (the externalization [D*]  of) an internal category D* in S.  
>>
However, for a Grothendieck topos S, this is the case on account of the  existence of a generating set.  An alternative construction of the stack  completion of a category in S is that of Andre Joyal and Myles Tierney, "Strong stacks and classifying spaces" in Category Theory, Proceedings of  Como 1990, LNM 1488, Springer, 213-236, 1991, by means of a Quillen model structure on Cat(S) whose weak equivalences are the weak equivalence functors.  This construction applies for instance to a Grothendieck topos S. In particular, for S Grothendieck topos, there is then an equivalence between the bicategory of anafunctors in Cat(S) and the Kleisli bicategory of the stack completion 2-monad on Cat(S).  
>>

Marta Bunge

************************************************
Marta Bunge
Professor Emerita
Dept of Mathematics and Statistics 
McGill UniversityBurnside Hall, Office 1005
805 Sherbrooke St. West
Montreal, QC, Canada H3A 2K6
Office: (514) 398-3810/3800  
Home: (514) 935-3618
marta.bunge@mcgill.ca 
http://www.math.mcgill.ca/~bunge/
************************************************



> Date: Mon, 31 Jan 2011 14:33:20 -0800
> Subject: Re: categories: Re: Fibrations in a 2-Category
> From: mshulman@ucsd.edu
> To: categories@mta.ca
> CC: marta.bunge@mcgill.ca; jean.benabou@wanadoo.fr; droberts@maths.adelaide.edu.au
> 
> Dear all,
> 
> In case there is any confusion, let me clarify that I have never
> claimed, myself, to have first invented/discovered the equivalence
> between the bicategories of anafunctors and of representable
> distributors.  I thought of it as a "folklore" sort of fact, hence why
> I did not attribute it, but Jean has pointed out that his email of 22
> Jan appears to be the first time anyone has written it down.
> 
> Best,
> Mike
  		 	   		  

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Fibrations in a 2-Category

[JeanBenabou]

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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Re: Fibrations in a 2-category

[JeanBenabou]

Dear David,

Thank you for your very kind offer. There are, at least, two reasons  
for which I'll have to think about it before I give an answer.

1- You say, I quote you:
"the input is no harder than writing in (La)TeX."
This seems to be very simple for you, but considering my very limited  
ability with computers (and this is an understatement) it will  
probably be almost impossible for me. For example I cannot type (La) 
TeX. If I need to have a text typed in TeX, I have to ask to a friend  
to type it for me. This is one of the reasons why I publish very  
little although I have many handwritten first drafts on various  
subjects in Category Theory.
2- Even assuming the friend would help me, the nLab is a wiki system,  
thus anybody would be able to modify my texts. And I'm sure I  
wouldn't like that. I'm not against discussion and I wouldn't object  
if one or many persons wrote their own texts, even if they are very  
critical of mine, provided they give mathematical arguments to  
justify their objections.

You might suggest solutions to 1 and 2, in which case I'd gladly  
accept your offer.

Thanks again, and best regards,

Jean

Le 13 janv. 11 à 02:37, David Roberts a écrit :

> On 12 January 2011 17:20, JeanBenabou <jean.benabou@wanadoo.fr> wrote:
>
>> Thus the official list does not permit such discussions. Can  
>> anybody tell me where they
>> can take place publicly?
>
> Dear Jean,
>
> you (and all other categories list readers) are welcome to add as much
> material on category theory of any sort as you see fit to the nLab.
>
> http://ncatlab.org/nlab/show/HomePage
>
> the input is no harder than writing in (La)TeX. For example, the page
>
> http://ncatlab.org/nlab/show/Grothendieck+fibration
>
> deals with fibrations from several different points of view, but if
> you see fit to expand it, I (and I assume others) would be very
> pleased.
>
> Or you could start some new topics at
>
> http://ncatlab.org/nlab/show/Jean+Benabou
>
> and I'm sure the nLab regulars will pitch in and lend a hand. As far
> as actively discussing these ideas go, there is the nForum
>
> http://www.math.ntnu.no/~stacey/Mathforge/nForum/
>
> where it is a simple matter to sign up. In all events, the discussions
> there are public and open for all to read.
>
> Best wishes, and good luck for what sounds like a very interesting  
> lecture,
>
> David Roberts


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Fibrations in a 2-category

[JeanBenabou]

I have seen very often the following "abstract" definition of a  
fibration in a 2-category C :
A map (i.e. a 1-cell) p: X --> S is a fibration iff for each object Y  
of C the functor C(Y,p):  C(Y,X) --> C(Y,S) is a fibration (in the  
usual sense) which depends "2-functorially" on Y.

Such an "obvious" definition is much too naive and does not give the  
correct notion in most examples.

1- Even if C= Cat, the 2-category of (small) categories, a fibration  
in the abstract sense is a Grothendieck fibration which admits a  
cleavage. Thus if we don't assume AC, which we don't need to define  
fibrations, it does not coincide with the usual one.

2- The situation is much worse in more general cases. Suppose E is a  
topos (this assumption is much too strong), and take C = Cat(E), the  
category of internal categories in E. On can define internal  
fibrations, and "fibrations" in the  previous "abstract" sense. They  
do not coincide.
It all boils down to the following remark: E and (E°, Set) are  
Toposes, the Yoneda functor E --> (E*,Set) preserves an reflects  
limits, but "nothing else" of the internal logic, which is needed to  
define internal fibrations.

Best to all,

Jean
   

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Re: Fibrations in a 2-category

[Ross Street]

Dear Jean

On 11/01/2011, at 6:31 PM, JeanBenabou wrote:

> 2- The situation is much worse in more general cases. Suppose E is a  
> topos (this assumption is much too strong), and take C = Cat(E), the  
> category of internal categories in E. On can define internal  
> fibrations, and "fibrations" in the  previous "abstract" sense. They  
> do not coincide.
> It all boils down to the following remark: E and (E°, Set) are  
> Toposes, the Yoneda functor E --> (E*,Set) preserves an reflects  
> limits, but "nothing else" of the internal logic, which is needed to  
> define internal fibrations.

I totally agree. An internal fibration between groups in a topos E is  
a group morphism whose underlying morphism in E is an epimorphism; for  
a representable fibration, it is a split epimorphism in E. Jack Duskin  
alerted me to this many years ago.

Never-the-less, the representable notion has had some uses. Actually,  
Dominic Verity and I also used representably Giraud-Conduché morphisms  
in

 	The comprehensive factorization and torsors, Theory and Applications  
of Categories 23(3) (2010) 42-75;

whereas there is an internal version (more generally applicable in the  
way you explain) of these too (in a topos, for example).

Have you written or published anything on these internal notions?

Best wishes,
Ross



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Re: Fibrations in a 2-category

[JeanBenabou]

Dear Ross,

Thank you for answering me and agreeing with me about the difference  
between internal fibrations and you call "representable" ones.I had  
started to write a few comments about what you said on the  
description of the Eilenberg- Moore category as sheaves on the  
Kleisli category for a generalized topology. I shall give a single  
answer to the two mails because they have some common features.
I apologize if this answer is very sketchy. Each of these questions  
would deserve a long development, which I can't give for two reasons.

1- Long mails are not accepted on the official list. (I don't want  
our moderator to think that any criticism is hinted at by this  
remark, but some questions, especially those dealing with  
"foundations", do need long developments if they are to be discussed  
seriously. Thus the official list does not permit such discussions.  
Can anybody tell me where they can take place publicly?)

2- I'm supposed to give a 2 hours lecture on Feb. 5 on "Transcendent  
" methods in Category Theory. The audience is quite "mixed":  
mathematicians, philosophers, linguists, and even... musicians! Quite  
a challenge since some of them have only the faintest notions about  
Category Theory. Thus most of my time and energy are devoted to its  
preparation.

§1- EILENBERG-MOORE VESUS KLEISLI

1.1.  I have no objection to the terminology; "sheaves for a  
generalized topoloogy". You could even drop "generalized" provided  
you indicate precisely what you mean by "topology" and "sheaf". After  
all Grothendieck did precisely that when he used these two words for  
his "topologies' on categories which were indeed "generalized" from  
"usual" topology.

1.2. There is an ambiguity in your mail when you write.

"After all, I believe Linton's work aimed at generalizing to all monads
the correspondence between monads of finite rank on Set  and
Lawvere theories, under which Eilenberg-Moore algebras become
product-preserving presheaves on part of the Kleisli category)"

As far as I remember Linton did not deal with "all" monads but with  
monads ON SETS.

The same ambiguity can be found e.g. in Lack's mail where he writes:

"Similarly, the Elienberg-Moore algebras can be seen as the  
presheaves on the Kleisli
category which send certain diagrams to limits."

He never mentions the fact that the monad is ON SETS. I suppose your  
"sheaf-interpretation" holds only for monads on Set, am I wrong?

What if we replace Set by another category, say S? I don't want to  
under estimate Lawvere's, or Linton's or your work, but; I apologize  
to theses authors this is a bit of "glorious past-time story",
Let me look at simple example, namely the "notion" of group.
In the case of sets there are many closely related notions, let me  
describe a few ones.
(i) The category Grp of groups which is monadic over Set
(ii)The Kleisli category of this monad
(iii) The Lawvere theory of groups, say Th(Grp)

If we replace Set by a category S with finite products, and denote by  
Grp(S) the category of internal groups of S; what is (part of ) the  
general picture?
(a) Grp(S) is the category of product preserving functors Th(Grp) -->  
S, which you can view as "S-valued sheaves on Th(Grp) for the obvious  
(generalized) topology . It is equipped with the forgetful functor U;  
Grp(S) --> S "evaluation at 1 (I apologize for such trivialities)
(b) Suppose U has a left adjoint F and let T be the associated monad.
(b.1) Is it obvious that Grp(S) is monadic for the monad T?
(b.2) What is the precise relation between Th(Grp) and the Kleisli  
category Kl(T) of T ?
(b.3) Can Grp(S) be interpreted as S valued sheaves on K(T) for a  
suitable topology?

A partial answer to these questions can be given when S is a topos  
with NNO, but,for me at least, even in that case, there remain many  
important questions which I can't answer. Has anybody been interested  
by the kind of questions raised in the previous subsection?

§2 FIBRATIONS AND "REPRESENTABLE" FIBRATIONS.

Thank you Ross for agreeing with me about the difference between  
(internal) fibrations and what you call "representable" ones.
Sorry if I disagree with you, but I tend to prefer the first ones.It  
is very easy to generalize important notions of Category Theory to 2- 
categories by making them "representable" but to me the real problem  
is to "internalize" these notions,( that is easy by using the  
internal language which I introduced for  precisely that purpose) and  
to STUDY THE PROPERTIES of these internalized notions; I am long past  
believing that ,apart from size conditions, ZF, with or without  
Universes or AC is enough to express all mathematical possibilities.  
As an,example is the important notion of "definability" which took me  
a long time to understand, by going "outside" of ZF

In your mail you say:

"An internal fibration between groups in a topos E is a group  
morphism whose underlying morphism in E is an epimorphism; for a  
representable fibration, it is a split epimorphism in E. Jack Duskin  
alerted me to this many years ago."

I do not know when Duskin "alerted" you. What I now is that I found  
the remark in the original paper of Grothendieck on fibrations (1961)  
and that I "internalized" it in 1970 when I introduced internal  
languages. I talked many times of this example in my seminar as an  
illustration of what internal languages could do. And Duskin attended  
my seminar for a whole year, and many other times for shorter periods  
But let's forget about this "detail" and concentrate about more  
important things.

For more than 20 years I have tried to convince people that  
fibrations and indexed categories are not "the same thing", even if  
we use AC and universes. For a long time I didn't convince you. I  
remember having offered 6 bottles of champagne to anyone who could  
prove, using only indexed categories, that the composite of two  
fibrations is a fibration. And I got an answer from you where you had  
to go through the Grothendieck construction for one of the indexed  
categories. Thus you didn't get the champagne. Of course, if you  
visit me in Paris, I'll be very glad to share with you a bottle, for  
old time's sake.
I contend that indexed categories are not the same even as cloven  
fibrations. They are the same IN SETS. But if you you go back to §1,  
you'll immediately see the difference. Fibrations and cloven  
fibrations can be internalized, e.g. in a topos (although this  
assumption is much too strong); Indexed categories cannot. And even  
if by some very complicated construction one could, in some special  
case, internalize them, unless you add some very strong artificial  
assumptions they would not coincide with internal fibrations, not  
even cloven ones.

Let me give a final trivial example to try again to convince you and  
a few other ones. Let me call for short "surjective" morphism of  
groups a morphism of groups such the the underlying morphism is an  
epi (better be a regular epi). Obviously they are stable under  
composition. How would you formulate this in terms of "internal  
indexed categories", assuming you have defined such notion?

There is a lot more I could say but the mail is already very long,  
and I hope it will be forwarded.
Thanks for reading me.

Best wishes,
Jean 
    

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Re: Fibrations in a 2-category

[Michael Shulman]

Dear Jean,

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.

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.  It can be represented by a
span  A <-- F --> B  whose left leg is fully faithful and surjective
on objects.  One intuition is that the objects of F over a\in A are
different "ways to compute a value" of the anafunctor at a.  Different
"ways to compute a value" may give different values, but they will be
canonically isomorphic.

For example, 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.

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.  Moreover, if we allow
morphisms between fibrations to be anafunctors as well, then the
bicategory of fibrations over C is biequivalent to the bicategory of
pseudofunctors C^{op} --> Cat_ana.  (This should not be read as saying
anything more than it says; in particular I would not claim that
fibrations are always "the same as" indexed categories even from this
viewpoint.  For fixed C, they form equivalent bicategories, which
makes them sufficiently "the same" for some purposes, but, as you have
pointed out, not for other purposes.)


Similarly, regarding "internalization," any ordinary (non-cloven)
fibration does give rise to an internal fibration in the bicategory
Cat_ana.  The same is true for internal fibrations and anafunctors in
a topos (the relevant "non-cartesian" parts of the internal logic of
the topos E having been incorporated into the definition of
Cat_ana(E)).  Unfortunately, since Cat_ana is only a bicategory, not a
strict 2-category, we do not get the strict notion of internal
fibration, but the weaker version as defined by Street, in which
cartesian liftings exist only up to isomorphism.  I think this is a
nice example of when one may be "forced" to use Street fibrations
rather than Grothendieck ones (never claiming, of course, that there
is anything necessarily "wrong" with Grothendieck fibrations when they
suffice).

For example, if p: P --> C is a (Grothendieck) fibration, f: A --> C
and g: A --> P are functors and m: f --> pg is a natural
transformation, then we can define an anafunctor A <-- H --> P in
which the objects of H are pairs (a,n), where a is an object of A and
n: x --> g(a) is a cartesian arrow in P with p(n) = m_a.  The functor
H --> A is surjective on objects because p is a fibration.  Then the
composite anafunctor ph: A --> C is naturally isomorphic to f, and
there is a natural transformation from h to g which lies over m
(modulo this isomorphism) and which is cartesian in Cat_ana(A,P) over
Cat_ana(A,C).  One can generalize to the case when f, g, and p are
also anafunctors and p is a Street fibration (suitably interpreted for
an anafunctor).

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.  As I mentioned above, many familiar
facts about category theory which normally use AC remain true without
it, if all notions are replaced by their corresponding "ana-"
versions.  Of course, this approach has the disadvantage that
anafunctors are more complicated than ordinary functors, and form a
bicategory rather than a strict 2-category; thus one may be forced
into using other weaker notions like Street fibrations, bilimits, etc.

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.
  However, many theorems which are true under AC now become false.  In
addition to the properties of fibrations as above, one also has to
distinguish between "having limits" in the sense of "every diagram has
a limit" versus the sense of "there is a function assigning a limit to
every diagram."

Personally, while there is nothing intrinsically wrong with (2), I
think (1) gives a more satisfactory theory.  It also has connections
to applications outside of category theory.  For instance, anafunctors
between internal categories in a topos are more or less equivalent to
morphisms between their stack completions, and in various parts of
mathematics internal categories, and notions equivalent to
anafunctors, are frequently used as representatives of stacks (Lie
groupoids, Hopf algebroids, moduli stacks, etc.).  So it is not just a
philosophical reason to prefer (1).  However, I respect that others
may disagree, and I'd be interested in hearing about mathematical
reasons to prefer (2).

Regards,
Mike



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Re: Fibrations in a 2-category

[Michal Przybylek]

On Fri, Jan 14, 2011 at 12:02 AM, Michael Shulman <mshulman@ucsd.edu> wrote:

> 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.

[...]

Interesting. But before I ask for references on ``anafunctors'' I
would like to know the following - is it false that for any (say)
topos T there exists a category C whose 2-category of internal
categories, functors, and natural transformations is (weakly)
equivalent to the bicategory Cat_ana(T)?


Best,
MRP


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Re: Fibrations in a 2-category

[David Roberts]

Hi Michal,

it is not *always* false. Consider the topoi Set and Set_choice, where
the first is the category of sets without choice and the second is
with choice. Then the bicategory of categories, anafunctors and
transformations in Set is equivalent (assuming choice in the
metalogic) to the 2-category of categories, functors and natural
transformations in Set_choice. This is (essentially) shown by Makkai
in his original anafunctors paper.

However, I doubt that it is always true (only a hunch). Also, one does
not need a topos as an ambient category in which to define
anafunctors, only a site where the Grothendieck pretopology is
subcanonical and singleton (single maps as covering families). The
topos case is when you take the regular pretopology.

And although you did not ask for a reference, here's one:

http://arxiv.org/abs/1101.2363

which builds on internal anafunctors introduced here

http://arxiv.org/abs/math.CT/0410328

and Makkai's original paper is available in parts from here:

http://www.math.mcgill.ca/makkai/anafun/

David

On 15 January 2011 09:14, Michal Przybylek <michal.przybylek@gmail.com> wrote:
> On Fri, Jan 14, 2011 at 12:02 AM, Michael Shulman <mshulman@ucsd.edu> wrote:
>
>> 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.
>
> [...]
>
> Interesting. But before I ask for references on ``anafunctors'' I
> would like to know the following - is it false that for any (say)
> topos T there exists a category C whose 2-category of internal
> categories, functors, and natural transformations is (weakly)
> equivalent to the bicategory Cat_ana(T)?
>
>
> Best,
> MRP


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Re: Fibrations in a 2-category

[David Roberts]

I was I think a bit hasty in my last post. I thought it was possible
to separate the use of Choice in the metalogic and in Set, but I can't
see how to stop Choice 'filtering down'. However if we work not with
categories of sets but more general categories, I can get a more
definite answer.


Let S be a site (with a subcanonical singleton pretopology) so that
the bicategory Cat_ana(S) is defined, and also assume that
coequalisers of reflexive pairs exist in S.

Theorem: If Cat_ana(S) is equivalent to a 2-category Cat(C) for some
category C with finite products and coequalisers of reflexive pairs,
then covers are split in S.

Proof:

  In what follows an _equivalence_ of bicategories is defined to be a
2-functor (weak or strict) which is essentially surjective and locally
fully faithful and essentially surjective. If one has Choice in the
metalogic, then one can find a 2-functor which is an inverse up to a
isotransformation etc.

Definition: Let B be a bicategory. An object x in B is called a
_discrete object_ if B(w,x) is equivalent to a set for all objects w.

Let do(B) denote the full sub-bicategory on the discrete objects. For
any object a in a category C there is a discrete object disc(a) in
Cat(C), and disc:C --> Cat(C) is a functor. There is also a  2-functor
Cat(S) --> Cat_ana(S) for a site S (with subcanonical singleton
pretopology), which is the identity on objects. Discrete objects in
Cat(S) are precisely discrete objects in Cat_ana(S).

Lemma: If B = Cat(C) for some category C with finite products and
coequalisers of reflexive pairs of arrows, then disc:C --> do(Cat(C))
is an equivalence.

Lemma: if B = Cat_ana(S) for some site S with coequalisers of
reflexive pairs of arrows, then disc:S --> do(Cat_ana(S)) is an
equivalence.

Lemma: Let F:B --> B' be an 2-functor. Then there is a 2-functor do(B)
--> do(B') (i.e. discrete objects are mapped to discrete objects). If
F is an equivalence then it reflects discrete objects.

Corollary: If F:B --> B' is an equivalence there is an equivalence
do(B) --> do(B') given by restriction of F.

So if we have an equivalence Cat(C) --> Cat_ana(S) and both C and S
satisfy the conditions of the first two lemmas, we have a co-span of
equivalences

C --> do(Cat_ana(S)) <-- S

Thus if one doesn't mind inverting equivalences as defined here, we
have an equivalence S --> C of categories.

Lemma: Given an equivalence of categories S --> C there is an
equivalence Cat(S) --> Cat(C).

Thus we have an equivalence Cat(S) --> Cat_ana(S). But this implies
that the appropriate version of internal Choice holds in S.      #


Going back to Michal's question, this would imply that in the topos S
all regular epimorphisms split, which is of course not always true.

David




On 17 January 2011 09:21, David Roberts <droberts@maths.adelaide.edu.au> wrote:
> Hi Michal,
>
> it is not *always* false. Consider the topoi Set and Set_choice, where
> the first is the category of sets without choice and the second is
> with choice. Then the bicategory of categories, anafunctors and
> transformations in Set is equivalent (assuming choice in the
> metalogic) to the 2-category of categories, functors and natural
> transformations in Set_choice. This is (essentially) shown by Makkai
> in his original anafunctors paper.
>
> However, I doubt that it is always true (only a hunch). Also, one does
> not need a topos as an ambient category in which to define
> anafunctors, only a site where the Grothendieck pretopology is
> subcanonical and singleton (single maps as covering families). The
> topos case is when you take the regular pretopology.
>
> And although you did not ask for a reference, here's one:
>
> http://arxiv.org/abs/1101.2363
>
> which builds on internal anafunctors introduced here
>
> http://arxiv.org/abs/math.CT/0410328
>
> and Makkai's original paper is available in parts from here:
>
> http://www.math.mcgill.ca/makkai/anafun/
>
> David
>

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Re: Fibrations in a 2-category

[Michael Shulman]

On Mon, Jan 17, 2011 at 1:02 AM, David Roberts
<droberts@maths.adelaide.edu.au> wrote:
> Theorem: If Cat_ana(S) is equivalent to a 2-category Cat(C) for some
> category C with finite products and coequalisers of reflexive pairs,
> then covers are split in S.

I don't think this argument quite works, but I think one can show
something almost as good, namely that if S is a topos, C has finite
limits, and Cat_ana(S) is equivalent to Cat(C), then S satisfies the
*internal* axiom of choice.

> Lemma: If B = Cat(C) for some category C with finite products and
> coequalisers of reflexive pairs of arrows, then disc:C --> do(Cat(C))
> is an equivalence.

This isn't quite right; a discrete object in Cat(C) is essentially an
internal equivalence relation in C, and so do(Cat(C)) is the category
of equivalence relations and functors between them (not morphisms
between their quotients).  This is a full subcategory of the free
exact completion C_{ex/lex}, which contains the free regular
completion C_{reg/lex} (the category of kernels).

We can say that disc:C --> do(Cat(C)) is fully faithful, and its
essential image consists of the projective objects in do(Cat(C))
(assuming that C has finite limits).  In particular, do(Cat(C)) has
enough projectives.

> Lemma: if B = Cat_ana(S) for some site S with coequalisers of
> reflexive pairs of arrows, then disc:S --> do(Cat_ana(S)) is an
> equivalence.

This I believe if S is exact, such as a topos.  The point is that an
effective equivalence relation in  S, regarded as an internal category
in S, admits a surjective weak equivalence to its quotient object,
regarded as a discrete internal category.  Thus, the two become
equivalent in Cat_ana(S), though not in general in Cat(S).

> Lemma: Let F:B --> B' be an 2-functor. Then there is a 2-functor do(B)
> --> do(B') (i.e. discrete objects are mapped to discrete objects). If
> F is an equivalence then it reflects discrete objects.

An equivalence of bicategories certainly preserves and reflects
discrete objects, which is all that matters for this proof.  (But a
general functor of bicategories need not preserve discrete objects.)

> So if we have an equivalence Cat(C) --> Cat_ana(S)

... then we can conclude that S, being equivalent to do(Cat_ana(S))
and hence to do(Cat(C)), has enough projectives, namely C.

Already this is a nontrivial restriction on a topos (or set-theoretic
axiom), although it can hold in the absence of IAC.  However, we can
say more.

First, if we identify C with the subcategory of projectives in S, then
the equivalence functor Cat(C) --> Cat_ana(S) must be, up to
equivalence, the inclusion which regards internal categories in C as
internal categories in S, and internal functors as internal
anafunctors.  For being an equivalence, it in particular preserves lax
codescent objects; but every internal category is a lax codescent
object formed of discrete internal categories, and the functor C -->
Cat(C) --> Cat_ana(S) is what we used to identify C with the
projective objects of S.

Therefore, since this functor is an equivalence, every internal
category in S must be equivalent, in Cat_ana(S), to an internal
category in C, i.e. an internal category in S formed of projective
objects.  Now for any object A of S, we have an internal category
1+A+1 \rightrightarrows 1+1 with "two objects" and A as the
object-of-morphisms from one to the other (and only identity arrows
otherwise).  If this category is equivalent in Cat_ana(S) to one
composed of projective objects, then we must have a surjective weak
equivalence to it from such a category, which is equivalent to giving
a well-supported projective object P such that PxPxA is projective.
Thus, any object A is "locally projective", which is sufficient for
IAC.

(If we are talking about set theoretical foundations, rather than
working in a topos, we could then pick an element p of P, which exists
since it is well-supported.  Then since the projectives are closed
under finite limits, the fiber of PxPxA over (p,p), namely A, would be
projective, and hence AC holds.)


I also think it's worth mentioning that if S merely has enough
projectives, then we can identify Cat_ana(S) (up to equivalence of
bicategories) with a full sub-2-category of Cat(S), consisting of
those internal categories whose object-of-objects is projective (but
with no condition on the object-of-morphisms).

In fact, such categories are the cofibrant objects in a model
structure on Cat(S), in which everything is fibrant and whose weak
equivalences are the internally fully-faithful and
essentially-surjective functors.  Thus, this is a particular case of
the fact that morphisms in the homotopy (2-)category of a model
category are represented by maps from a cofibrant replacement to a
fibrant replacement.

(When S is a Grothendieck topos, there is also a model structure on
Cat(S) with those weak equivalences in which every object is
cofibrant, and in which the fibrant objects are stacks.  I believe
this was proven by Joyal and Tierney in their paper "Strong stacks and
classifying spaces".)

The set-theoretic axiom that "there exist enough projective sets" is a
weak form of choice called the "presentation axiom" or "COSHEP" ("the
Category Of Sets Has Enough Projectives").  It implies dependent
choice and some other weak forms of choice, and tends to hold in
models arising from type theory.  So if one is willing to accept that
axiom in lieu of full AC, or one is working in a topos that has enough
projectives (such as, notably, the effective topos), then one can
avoid talking about anafunctors by restricting to internal categories
with projective object-of-objects.  I don't know whether there is a
dual set-theoretic "axiom of small stack completions".

Regards,
Mike


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