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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Jean On 12/01/2011, at 5:50 PM, JeanBenabou wrote:
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)"
I agree there is ambiguity. Linton's paper is in a Lecture Note volume now in TAC Reprints for people to see.
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?
The comment about generalized topology was meant to be informal. The Proposition it followed gave a Yoneda structure version of Linton's pullback of the Yoneda embedding making the Eilenberg-Moore category a full subcategory of presheaves on the Kleisli category. This works for monads on any category.
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",
Yes, we are talking about "the good old days". But that is what the original question was about.
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?
Yes, I have wondered about some of those matters. There are many questions I cannot answer.
§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.
I am not sure what you are disagreeing about. Are you disagreeing that I agree with you? I certainly believe you prefer the first ones. I would like to hear what you have to say about them. I had something to say about the representably-defined ones but believe the others are more important. You are looking at harder problems by studying them. At no time have I claimed that the problems I look at are the most important ones.
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 was not fortunate enough to be around Grothendieck or to attend your 1970 lectures. So Jack Duskin was able to tell me some aspects of what he had heard from you. Similarly I didn't originally learn the group epimorphism example from reading Grothendieck; I saw it in Gray's LaJolla paper if my memory serves me correctly. Gray refers to Grothendieck whom I read later. I am glad that you have pointed out that you had lectured on the internal version of this in 1970.
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.
If you recall what I said at the time was that I would not bid for your champagne prize. I was not offering an answer. The reason is that it cannot be done. Fibrations cannot be composed using homomorphisms of bicategories from a category into Cat. As you know, the question doesn't make sense. However, homomorphisms (and morphisms) of bicategories are also useful concepts and I thank you for giving them to us. I wish you the best for your lecture preparation. I am sure the lectures will be inspiring. Best wishes, Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
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 -- 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/ ]
participants (5)
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David Roberts -
JeanBenabou -
Michael Shulman -
Michal Przybylek -
Ross Street