Fibrations in a 2-category

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: Re: Fibrations in a 2-category

[David Roberts]

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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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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