Re: Composition of Fibrations

Re: Composition of Fibrations

[Jean Bénabou]

A few weeks ago there has been a discussion about stability by composition of fibrations, bifibrations, and similar notions. Obviously the results depend on how such notions are defined.  I would like to make a few comments, in particular about Steve Vickers' mail, since all the other participants to the discussion seemed to accept his approach.

(I )    VICKERS' DEFINITION OF FIBRATION
if C is a 2-category with comma objects and 2-pullbacks, a one cell p: B -> A is a fibration iff it satisfies the Chevalley condition.

Let us test this definition in special cases.
If  S is a category with finite limits the 2-category Cat(S) of internal categories in S satisfies Vickers' conditions hence we know when an internal functor is a fibration. 
Let Set be the category of sets, except WE DON'T ASSUME THE AXIOM OF CHOICE (AC). Then Cat(Set), abbreviated by Cat, is the 2-Category of small categories.
An easy verification shows that a functor p: B -> A  satisfies the Chevalley condition iff it is a fibration which admits a cleavage. Thus Vickers' argument, in that case, gives as result: fibrations WHICH ADMIT A CLEAVAGE are stable by composition. 
On the other hand, it is easy to show that: Every small fibration has a cleavage is equivalent to AC. This well known fact can be very much strengthened by the following example:

If AC does not hold in Set, one can construct in Cat a bifibration  p: B -> A  with internal products and coproducts where A and B are pre-ordered sets, with pullbacks preserved by p, every map of B is both cartesian and cocartesian, and add each of the following conditions:
(i)  p has neither a cleavage nor a cocleavage. 
(ii) A bit surprisingly:   p is a split fibration but has no cocleavage.
(iii) Dual of (ii):   p is a cosplit cofibration but has no cleavage.

And of course we don't need AC to show that arbitrary fibrations in Cat are stable by composition.

(II)     2-CTEGORICAL FIBRATIONS
Other definitions of fibrations in an arbitrary 2-Category C have been proposed. The principal one, based on Yoneda, is:  A one cell  p: B -> A of  C  is a fibration iff for every object X of C the obvious functor  C(X,B) -> C(X,A) is a fibration in Cat, functorial in X. 
If C has comma objects and 2-pullbacks, it is easy to see that this is equivalent to Vickers' notion, and we have already seen how it can be inadequate.
Of course, I don't refer here to Sreet's notion which describes a totally different kind of fibration, stable by equivalences.

(III)    BIFIBRATIONS
For bifibrations the situation is even more confusing: Ghani defines them, in Cat, by the existence of left adjoints to the reindexing functors, Except that without AC reindexing functors need not exist. Vickers uses two duals of the 2-category C where the fibration lives. However if C has comma objects and 2-pullbacks, there is no reason why these duals have the same properties. Moreover, even in Cat, Vickers' approach will work only for bifibrations which have both a cleavage and a co-cleavage. 
Thus the wide generalization asserted by Vickers imposes in the well known situations drastic and unnecessary restrictions. (Compare with the example at the end of (I))


(IV)    INTERNAL FIBRATIONS. 
For a long time I have insisted on the fact that the the theory of fibrations is first order and can be internalized. In particular in Cat(S) where S is a topos. 
Let me give a very simple example. Suppose A and B are groups of S and  p: B -> A  is a group morphism. Then p is an internal fibration iff it is an epi of S. It satisfies the Chevalley condition iff p admits a splitting in S.. 
Without any such splitting, internally, every element of B is an iso, hence it is both cartesian and co-cartesian. Thus p is a bifibration. Moreover, again internally, since A and B are groups, every commutative  square of B, or A, is a pullback, thus B and A have pullbacks preserved by p, and in B the pullback of a cocartesian map along a cartesian map exists and is cocartesian. Hence p is a fibration with internal sums, it has also trivially internal products. 
But it has neither cleavages, cocleavages, nor reindexings. 
What would the theory of groups, torsors, classifiers etc. in a topos S  look like if we were forced to assume that S satisfies AC ?

Let me add that the internalization works not only for fibrations, but also for prefibrations and even for the (pre)foliations which I have defined and studied.

(V)    THE GROTHENDIECK CONSTRUCTION AND INDEXED CATEGORIES.
Suppose S is a topos. If S satisfies AC, every fibration in Cat(S) will have a cleavage. 
However, even if S = Set , the Grothendieck construction doesn't make sense without further assumptions, because:  if A is an internal category the very notion of a pseudo functor from A(op) to Cat(S) does not make sense since A is internal and Cat(S) is not an internal 2-category (a notion which can be easily defined)

One of the mottos of the Elephant is that fibrations and indexed categories are essentially equivalent, but the second notion doesn't even make sense. it is defined as a pseudo functor from a category, which is a mathematical object, into the meta 2-category Cat. Moreover, to get the 2-category of indexed categories, we are required to collect ALL such pseudo functors and their transformations. Thus I ask the question: What is such a collection, a meta-meta 2-category?

Yet another example:  If we use Lawvere's Category of categories as a foundation for mathematics, fibrations, or cloven fibrations, make perfect sense, but indexed categories don't, let alone the equivalence between the two notions.

In the introduction of the Elephant one can read, I quote:
We should make the smallest possible demands on the metatheory within which we interpret the theory of categories (and in particular we shall not assume that it satisfies any form of the axiom of choice ...
After reading carefully the chapter on indexed categories and fibrations, I ask Peter Johnstone if the following assertion would not be be more appropriate:
We shall make, in particular in Chapter B1, the greatest possible demands on the metatheory and in particular assume that it satisfies the strongest form of the axiom of choice..

Incidentally, that is exactly what Grothendieck does: He uses the axiom of universes, and the tau symbol which is the strongest possible form of AC.
But, even under such strong assumptions, I think he would object (and so would many other persons), if only for aesthetic reasons, to the following sentence which can be found many times in the Elephant:
Let  p be a fibration and  C be THE associated indexed category, ...
And this of course, according to Johnstone, without ANY form of AC. 

I'd have many more comments but this mail is already a bit long. I apologize for this length, and also for using capital letters in many places where italics or quotation marks would have been more appropriate. But ... HTML oblige. 




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Re: Composition of Fibrations

[Steve Vickers]

Dear Jean,

Thank you for your detailed comments.

Something I should say straight away is that the duality argument I had in mind, dualizing 2-cells, might be OK to deal with left adjoints to reindexing  but was completely wrong for right adjoints. Already, Richard Garner and Claudio Ermida (thanks to both of them) have shown me that it doesn't do the job.

I also want to stress that at no point did I intend to set up my own definition of fibration. I was following Street's "Fibrations and Yoneda's lemma in  a 2-category", which defines fibrations as those 1-cells that carry pseudoalgebra structure for a certain 2-monad, and then proves (Proposition 9) that  this is equivalent to what Street refers to as the Chevalley condition. If "Vickers' definition" is not equivalent to that then I have made a mistake somewhere. 

Have you found a discrepancy between the "Vickers definition" and Street? At  one point you write "Of course, I don't refer here to Street's notion which  describes a totally different kind of fibration, stable by equivalences."

I agree that the concept I have been using includes cleavage (and, for a bifibration, cocleavage). I cannot assume AC in what I do, and I rather imagined that structure something like the Chevalley criterion was needed in order to deal with its absence. However, I admit I am not so familiar with the fully general notion of fibration. For me the Chevalley condition seemed enough to do what I needed in the 2-category Loc of locales and my remarks were based on that experience.

Best wishes,

Steve.



> On 20 Jul 2014, at 17:18, Jean Bénabou <jean.benabou@wanadoo.fr> wrote:
> 
> A few weeks ago there has been a discussion about stability by composition  of fibrations, bifibrations, and similar notions. Obviously the results depend on how such notions are defined.  I would like to make a few comments, in particular about Steve Vickers' mail, since all the other participants to the discussion seemed to accept his approach.
> 
> (I )    VICKERS' DEFINITION OF FIBRATION
> if C is a 2-category with comma objects and 2-pullbacks, a one cell p: B -> A is a fibration iff it satisfies the Chevalley condition.
> 
> Let us test this definition in special cases.
> If  S is a category with finite limits the 2-category Cat(S) of internal categories in S satisfies Vickers' conditions hence we know when an internal functor is a fibration. 
> Let Set be the category of sets, except WE DON'T ASSUME THE AXIOM OF CHOICE (AC). Then Cat(Set), abbreviated by Cat, is the 2-Category of small categories.
> An easy verification shows that a functor p: B -> A  satisfies the Chevalley condition iff it is a fibration which admits a cleavage. Thus Vickers' argument, in that case, gives as result: fibrations WHICH ADMIT A CLEAVAGE are  stable by composition. 
> On the other hand, it is easy to show that: Every small fibration has a cleavage is equivalent to AC. This well known fact can be very much strengthened by the following example:
> 
> If AC does not hold in Set, one can construct in Cat a bifibration  p: B -> A  with internal products and coproducts where A and B are pre-ordered sets, with pullbacks preserved by p, every map of B is both cartesian and cocartesian, and add each of the following conditions:
> (i)  p has neither a cleavage nor a cocleavage. 
> (ii) A bit surprisingly:   p is a split fibration but has no cocleavage.
> (iii) Dual of (ii):   p is a cosplit cofibration but has no cleavage.
> 
> And of course we don't need AC to show that arbitrary fibrations in Cat are stable by composition.
> 

...


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Re: Composition of Fibrations

[Jean Bénabou]

Dear Steve,

Thank you for your prompt  answer. 

Let me first clarify a possible ambiguity. 
The Street fibrations I was referring to are defined in his paper: 
Fibrations in bicategories.  Cahiers Top. Geom. Diff. 21 (1980)
When the bicategory is Cat, they do not coincide with the usual fibrations. In particular every equivalence is a Street fibration.

There might be another ambiguity about what you call the Chevalley criterium. Could you please tell me with precision what it is (I assume p: B -> A is a map in a 2-category C with comma objects and 2-pullbacks)

I shall come back to mathematical questions as soon as these two ambiguities are solved.

Best wishes,

Jean


Le 21 .. 2014 à 14:30, Steve Vickers a écrit :

> Dear Jean,
> 
> Thank you for your detailed comments.
> 
> Something I should say straight away is that the duality argument I had in mind, dualizing 2-cells, might be OK to deal with left adjoints to reindexing but was completely wrong for right adjoints. Already, Richard Garner and Claudio Ermida (thanks to both of them) have shown me that it doesn't do the job.
> 
> I also want to stress that at no point did I intend to set up my own definition of fibration. I was following Street's "Fibrations and Yoneda's lemma in a 2-category", which defines fibrations as those 1-cells that carry pseudoalgebra structure for a certain 2-monad, and then proves (Proposition 9) that this is equivalent to what Street refers to as the Chevalley condition. If "Vickers' definition" is not equivalent to that then I have made a mistake somewhere. 
> 
> Have you found a discrepancy between the "Vickers definition" and Street? At one point you write "Of course, I don't refer here to Street's notion which describes a totally different kind of fibration, stable by equivalences."
> 
> I agree that the concept I have been using includes cleavage (and, for a bifibration, cocleavage). I cannot assume AC in what I do, and I rather imagined that structure something like the Chevalley criterion was needed in order to deal with its absence. However, I admit I am not so familiar with the fully general notion of fibration. For me the Chevalley condition seemed enough to do what I needed in the 2-category Loc of locales and my remarks were based on that experience.
> 
> Best wishes,
> 
> Steve.
> 
> 
> 

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Re: Composition of Fibrations

[Steve Vickers]

Dear Jean,

Street's result is as follows. The arrow p: E -> B is a 0-fibration over B if and only if the arrow
   p~ : ΦE -> p/B
corresponding to the 2-cell

ΦE  --pd1--> B
|                      ||
d0                   ||
|         pλ  =>   ||
v                     ||
E   --p------> B

has a left adjoint with unit an isomorphism.

Here ΦE = E/E and p/B are comma objects, d0 and d1 are projections, and λ is the canonical 2-cell in a comma square (in this case for ΦE). 0-fibration is opfibration.

Regards,

Steve.

> On 21 Jul 2014, at 19:02, Jean Bénabou <jean.benabou@wanadoo.fr> wrote:
> 
> Dear Steve,
> 
> Thank you for your prompt  answer. 
> 
> Let me first clarify a possible ambiguity. 
> The Street fibrations I was referring to are defined in his paper: 
> Fibrations in bicategories.  Cahiers Top. Geom. Diff. 21 (1980)
> When the bicategory is Cat, they do not coincide with the usual fibrations. In particular every equivalence is a Street fibration.
> 
> There might be another ambiguity about what you call the Chevalley criterium. Could you please tell me with precision what it is (I assume p: B -> A is a map in a 2-category C with comma objects and 2-pullbacks)
> 
> I shall come back to mathematical questions as soon as these two ambiguities are solved.
> 
> Best wishes,
> 
> Jean
> 

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Re: Composition of Fibrations

[Jean Bénabou]

Dear Steve,

At least one ambiguity is solved. Chevalley gave as criterium (for opfibrations) that the arrow 
p~ : ΦE -> p/B  in your mail has a left adjoint with unit the identity.
When the 2-category is Cat this condition is satisfied iff  p is an opfibration which has an opcleavage. The choice of the adjoint defines the opcleavage.

Let us for the sake of precision call  Street criterium the existence of a left adjoint with unit an iso, and Street opfibrations (in Cat) the functors which satisfy this condition. 
They need not be opfibrations in the sense of Grothendieck which is almost unanimously adopted. It is unfortunate to have given them the name of (op)fibrations, not only because of the ambiguity as we have seen, but because the fibers are meaningless, in particular the fibers over two isomorphic objects of the base B need not be isomorphic. 

I'm almost sure that Neil Ghani, Richard Garner, Claudio Hermida and Thomas Streicher meant Grothendieck fibrations, and the genuine Chevalley condition in your answer, as I did.

Regards,

Jean

 

> Dear Jean,
> 
> Street's result is as follows. The arrow p: E -> B is a 0-fibration over B if and only if the arrow
>  p~ : ΦE -> p/B
> corresponding to the 2-cell
> 
> ΦE  --pd1--> B
> |                      ||
> d0                   ||
> |         pλ  =>   ||
> v                     ||
> E   --p------> B
> 
> has a left adjoint with unit an isomorphism.
> 
> Here ΦE = E/E and p/B are comma objects, d0 and d1 are projections, and λ is the canonical 2-cell in a comma square (in this case for ΦE). 0-fibration is opfibration.
> 
> Regards,
> 




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Re: Composition of Fibrations

[Steve Vickers]

[Ross: Any comments?]

Dear Jean,

I wasn't properly aware of those issues around choosing iso or equality, so it's lucky I got into this discussion. My intuition of what you are saying is  that with iso, roughly speaking, the reindexing is only pseudofunctorial. For example, if you pull back along the diagonal B -> ΦB, to get the reindexing along identities, then you get an endofunctor of each fibre that is isomorphic to the identity. Am I on the right lines? Is this all written down  somewhere?

After your messages I noticed that Street has a remark after his proposition, whose significance I overlooked:

"Compare the above proposition with Gray [2] p.56; so we have related the definition of 0-fibration here with the definition of opfibration in [2] when K  = Cat. Notice that the unit of the adjunction l -| p~ for Gray is not just an isomorphism but an identity. It is worth pointing out the reason for this since we will need the observation in the next paper. A 0-fibration will be called normal when there is a normalized pseudo L-algebra structure on it.  In Cat every 0-fibration is normal, but in other categories this need not be the case. In the proof of the Chevalley criterion, if ζ is an identity then so is η. So, for a normal 0-fibration,
p~ : ΦE -> p/B has a left adjoint with unit an identity."

Notes:
1. Gray [2] = "Fibred and cofibred categories", La Jolla.
2. I don't know which paper Street means by "the next paper".
3. In a pseudo L-algebra E, with structure morphism c: LE -> E, ζ denotes the isomorphism from Id_E to unit composed with c. E is normalized if ζ is equality.
4. η is the unit of the adjunction.

But that seems to claim that in Cat it doesn't matter whether you use iso or  equality in the Chevalley condition. Does that accord with your understanding?

Regards,

Steve.

> On 22 Jul 2014, at 05:24, Jean Bénabou <jean.benabou@wanadoo.fr> wrote:
> 
> Dear Steve,
> 
> At least one ambiguity is solved. Chevalley gave as criterium (for opfibrations) that the arrow 
> p~ : ΦE -> p/B  in your mail has a left adjoint with unit the identity.
> When the 2-category is Cat this condition is satisfied iff  p is an opfibration which has an opcleavage. The choice of the adjoint defines the opcleavage.
> 
> Let us for the sake of precision call  Street criterium the existence of a  left adjoint with unit an iso, and Street opfibrations (in Cat) the functors which satisfy this condition. 
> They need not be opfibrations in the sense of Grothendieck which is almost  unanimously adopted. It is unfortunate to have given them the name of (op)fibrations, not only because of the ambiguity as we have seen, but because the fibers are meaningless, in particular the fibers over two isomorphic objects of the base B need not be isomorphic. 
> 
> I'm almost sure that Neil Ghani, Richard Garner, Claudio Hermida and Thomas Streicher meant Grothendieck fibrations, and the genuine Chevalley condition in your answer, as I did.
> 
> Regards,
> 
> Jean
> 
> 
> 
>> Dear Jean,
>> 
>> Street's result is as follows. The arrow p: E -> B is a 0-fibration over B  if and only if the arrow
>> p~ : ΦE -> p/B
>> corresponding to the 2-cell
>> 
>> ΦE  --pd1--> B
>> |                      ||
>> d0                   ||
>> |         pλ  =>   ||
>> v                     ||
>> E   --p------> B
>> 
>> has a left adjoint with unit an isomorphism.
>> 
>> Here ΦE = E/E and p/B are comma objects, d0 and d1 are projections, and λ is the canonical 2-cell in a comma square (in this case for ΦE). 0-fibration is opfibration.
>> 
>> Regards,
> 
> 


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Re: Composition of Fibrations

[Ross Street]

Dear Steve and Jean

On 23 Jul 2014, at 12:55 am, Steve Vickers <s.j.vickers@cs.bham.ac.uk> wrote:

> In Cat every 0-fibration is normal,

This statement in SLNM420 is false (since in that paper 0-fibration 
means existence of a pseudo-L-algebra structure). 
I must have temporarily believed it when I wrote it.
I was using ``normal’’ to mean strict unit condition.
Grothendieck fibrations with chosen cleavage 
are the normal pseudo-L-algebras.

I apologize profusely for confusion caused.

Ross


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Re: Composition of Fibrations

[Eduardo J. Dubuc]

I always have present that Grothendieck himself explicitly discarded his
notion of "cat?gorie cliv?e" (ie indexed category) introduced in the
S?minaire Bourbaki (1959) in favor of his notion of "cat?gorie fibr?e"
introduced in SGA1 (1961) (see SLN Vol 224 remark page 193).


On 20/07/14 13:18, Jean B?nabou wrote:
> A few weeks ago there has been a discussion about stability by composition of fibrations, bifibrations, and similar notions. Obviously the results depend on how such notions are defined.  I would like to make a few comments, in particular about Steve Vickers' mail, since all the other participants to the discussion seemed to accept his approach.
>
> (I )    VICKERS' DEFINITION OF FIBRATION
> if C is a 2-category with comma objects and 2-pullbacks, a one cell p: B -> A is a fibration iff it satisfies the Chevalley condition.
>
> Let us test this definition in special cases.
> If  S is a category with finite limits the 2-category Cat(S) of internal categories in S satisfies Vickers' conditions hence we know when an internal functor is a fibration.
> Let Set be the category of sets, except WE DON'T ASSUME THE AXIOM OF CHOICE (AC). Then Cat(Set), abbreviated by Cat, is the 2-Category of small categories.
> An easy verification shows that a functor p: B -> A  satisfies the Chevalley condition iff it is a fibration which admits a cleavage. Thus Vickers' argument, in that case, gives as result: fibrations WHICH ADMIT A CLEAVAGE are stable by composition.
> On the other hand, it is easy to show that: Every small fibration has a cleavage is equivalent to AC. This well known fact can be very much strengthened by the following example:
>
> If AC does not hold in Set, one can construct in Cat a bifibration  p: B -> A  with internal products and coproducts where A and B are pre-ordered sets, with pullbacks preserved by p, every map of B is both cartesian and cocartesian, and add each of the following conditions:
> (i)  p has neither a cleavage nor a cocleavage.
> (ii) A bit surprisingly:   p is a split fibration but has no cocleavage.
> (iii) Dual of (ii):   p is a cosplit cofibration but has no cleavage.
>
> And of course we don't need AC to show that arbitrary fibrations in Cat are stable by composition.
>
> (II)     2-CTEGORICAL FIBRATIONS
> Other definitions of fibrations in an arbitrary 2-Category C have been proposed. The principal one, based on Yoneda, is:  A one cell  p: B -> A of  C  is a fibration iff for every object X of C the obvious functor  C(X,B) -> C(X,A) is a fibration in Cat, functorial in X.
> If C has comma objects and 2-pullbacks, it is easy to see that this is equivalent to Vickers' notion, and we have already seen how it can be inadequate.
> Of course, I don't refer here to Sreet's notion which describes a totally different kind of fibration, stable by equivalences.
>
> (III)    BIFIBRATIONS
> For bifibrations the situation is even more confusing: Ghani defines them, in Cat, by the existence of left adjoints to the reindexing functors, Except that without AC reindexing functors need not exist. Vickers uses two duals of the 2-category C where the fibration lives. However if C has comma objects and 2-pullbacks, there is no reason why these duals have the same properties. Moreover, even in Cat, Vickers' approach will work only for bifibrations which have both a cleavage and a co-cleavage.
> Thus the wide generalization asserted by Vickers imposes in the well known situations drastic and unnecessary restrictions. (Compare with the example at the end of (I))
>

...


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cleavages and choice

[Thomas Streicher]

Dear Jean,

of course, you are right when emphasizing that one need choice for
classes to endow an "anonymous" fibration with a cleavage.
But that applies also to catgeories with say binary products. One
needs choice for classes in order to choose a product cone for every
pair of objects.
In many instances, however, categories come together with a choice of
products and fibrations come together with a choice of a cleavage.

For example Set comes with a choice of a cleavage. Fibrations arising
from internal categories are even split. Many constructions on fibrations
allow one to choose a cleavage given cleavages for the arguments.
Do you know of any construction on fibrations which is not "cleavage
preserving" in this sense?

Of course, one should not require cartesian functors to preserve
cleavages just as one should not require functors to preserve chosen
products.

Thomas


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Re: cleavages and choice

[Jean Bénabou]

Dear Thomas,

I am a bit surprised that you, of all people, should defend cleavages, i.e. indexed categories. As far as I remember, there not many of them in the notes you wrote on Fibered categories a la Benabou.
I remind you that these notes were written not only after lectures I gave, but after long conversations we had, many times, in my flat, and also a few days you spent in my house in the south of France, where, for at least 10 hours everyday I explained in detail to you my work on fibered categories and corrected many mistakes you made in first drafts  of that paper.
Nevertheless, for the sake of the people on the category list,to whom this message is also addressed, I shall answer your questions an remarks.

Let  p: X --> S be a surjective group homomorphism. It is a fibration, and a cleavage is a section of p (in Set). This was explicitly noted by Grothendieck more than 50 years ago! 
Does such a  p come equipped with a cleavage? Take for p the morphim :  R --> R/Z  of the reals on the circle. Suppose I were to teach periodic functions, i.e functions with domain R which factor through R/Z. Wouldn't it be ridiculous to use a section of p ? Which one by the way?

Take the theorem : The composite of two fibrations is a fibration. Does it need cleavages, i.e. AC for classes, to be proved? Of course, if you are cleavage fan, as you seem to be, you can add that, given cleavages of p and q one gets an associated cleavage of  pq.

Let's look at an important example,namely categories S with pull backs, not with choice of pull backs mind you. This is a first order notion, saying that : for every cospan of S there exists a universal span making the obvious square commutative.
Then, without AC, you can prove that the functor  Codom: S^2 --> S  is a fibration. And again, if you are cleavage happy, add that a cleavage of this fibration, if it exists, (I'm not assuming AC)  is a choice of pullbacks in S.

I could multiply the examples. But let's look at an important question. Suppose you prove an intrinsic result about fibrations, using cleavages, in principle you'd have to see what happens when you change cleavages. And don't wave your hands and tell me that, for formal 2-categorical reasons, the result is obvious. I'll believe you only when you write a precise metatheorem which covers ALL the cases.
You are convinced, and I am convinced, and everybody is convinced that such a metatheorem is not necessary. But that is NOT A PROOF !  And why are you convinced? Because, even if you say the contrary, deep in your mind you KNOW that intrinsic properties of fibrations AND cartesian functors should not refer to cleavages. Let me insist on the fact that the mythic metatheorem should also cover cartesian functors  F: X --> X' where you change the cleavages of both X and X'

Of course the theorem I mentioned in my mail on pre foliations, applies to fibrations and gives new results in that case. But this theorem is true for  F: X --> X'  where X is a prefoliation hence, even with AC, has no cleavage, and X' is an arbitrary category over S, i.e. has even less cleavages than X.

In order not to make this mail too long I have not, but I should have, mentioned internalization where cleavages are even more problematic.

Best to all,
Jean


Le 30 juil. 2014 à 17:06, Thomas Streicher a écrit :

> Dear Jean,
> 
> of course, you are right when emphasizing that one need choice for
> classes to endow an "anonymous" fibration with a cleavage.
> But that applies also to catgeories with say binary products. One
> needs choice for classes in order to choose a product cone for every
> pair of objects.
> In many instances, however, categories come together with a choice of
> products and fibrations come together with a choice of a cleavage. 
> 
> For example Set comes with a choice of a cleavage. Fibrations arising
> from internal categories are even split. Many constructions on fibrations 
> allow one to choose a cleavage given cleavages for the arguments.
> Do you know of any construction on fibrations which is not "cleavage
> preserving" in this sense?
> 
> Of course, one should not require cartesian functors to preserve
> cleavages just as one should not require functors to preserve chosen
> products.
> 
> Thomas



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Re: cleavages and choice

[Eduardo J. Dubuc]

Dear all,

I prefer fibrations over fibrations furnished with a cleavage (indexed
categories) many times for reasons purely pragmatical mixed with an
aesthetic philosophy.

Suppose you are dealing with fibrations where a canonical cleavage is
present, suppose even that these cleavages come first and that the
fibration is just a  conceptual context around them. Even in this case,
faced to the need to produce a proof, if you succeed to find one without
utilizing the cleavages, you will have something much nicer than the
cleavage arguing. It will also give you a deeper understanding and a
truthful light on the situation.

Suppose you do not care about foundations, axiom of choice, or things of
that sort. You should still prefer fibrations. They are simpler, more to
the point, and CERTAINLY AHEAD IN THE PROGRESS OF MATHEMATICS.


NOTE: I wonder why so many people are so happy working with pull-backs
and pull-back preserving functors (*) without even thinking in
introducing a choice of pullbacks, and when it comes to fibrations, feel
the need to introduce and work with cleavages.

(*) for example even when dealing with the category of sets (or
categories whose objects have an underlying set), which are plenty of
choices of pull-backs, for example, inverse image of a subset, the
standard construction as a subset of the set of pairs, etc. We precisely
teach in category theory courses that you should not work with any
particular choice between the choices.

We all agree that it is neither necessary not good to choose a choice
between all possible choices. This is precisely the progress that
represents category theory thinking over set theory thinking. See for
example, in the dawn of category theory, the considerations of Mac Lane
concerning the fact that a quotient of a quotient of a group is not a
quotient (as it is still now taught in algebra courses, category theory
thinking has not arrived there yet).

You may say that the choice of a cleavage is at a different level than
all this, but, essentially, deep down, for me it is the same. There is
an old way of thinking (as Grothendieck said, SLN224, page 193) that
hesitates in face of fibrations and prefer to work with a chosen cleavage.



On 30/07/14 14:56, Jean B?nabou wrote:
> Dear Thomas,
>
> I am a bit surprised that you, of all people, should defend cleavages, i.e. indexed categories. As far as I remember, there not many of them in the notes you wrote on Fibered categories a la Benabou.
> I remind you that these notes were written not only after lectures I gave, but after long conversations we had, many times, in my flat, and also a few days you spent in my house in the south of France, where, for at least 10 hours everyday I explained in detail to you my work on fibered categories and corrected many mistakes you made in first drafts  of that paper.
> Nevertheless, for the sake of the people on the category list,to whom this message is also addressed, I shall answer your questions an remarks.
>
> Let  p: X --> S be a surjective group homomorphism. It is a fibration, and a cleavage is a section of p (in Set). This was explicitly noted by Grothendieck more than 50 years ago!
> Does such a  p come equipped with a cleavage? Take for p the morphim :  R --> R/Z  of the reals on the circle. Suppose I were to teach periodic functions, i.e functions with domain R which factor through R/Z. Wouldn't it be ridiculous to use a section of p ? Which one by the way?
>
> Take the theorem : The composite of two fibrations is a fibration. Does it need cleavages, i.e. AC for classes, to be proved? Of course, if you are cleavage fan, as you seem to be, you can add that, given cleavages of p and q one gets an associated cleavage of  pq.
>
> Let's look at an important example,namely categories S with pull backs, not with choice of pull backs mind you. This is a first order notion, saying that : for every cospan of S there exists a universal span making the obvious square commutative.
> Then, without AC, you can prove that the functor  Codom: S^2 --> S  is a fibration. And again, if you are cleavage happy, add that a cleavage of this fibration, if it exists, (I'm not assuming AC)  is a choice of pullbacks in S.
>
> I could multiply the examples. But let's look at an important question. Suppose you prove an intrinsic result about fibrations, using cleavages, in principle you'd have to see what happens when you change cleavages. And don't wave your hands and tell me that, for formal 2-categorical reasons, the result is obvious. I'll believe you only when you write a precise metatheorem which covers ALL the cases.
> You are convinced, and I am convinced, and everybody is convinced that such a metatheorem is not necessary. But that is NOT A PROOF !  And why are you convinced? Because, even if you say the contrary, deep in your mind you KNOW that intrinsic properties of fibrations AND cartesian functors should not refer to cleavages. Let me insist on the fact that the mythic metatheorem should also cover cartesian functors  F: X --> X' where you change the cleavages of both X and X'
>
> Of course the theorem I mentioned in my mail on pre foliations, applies to fibrations and gives new results in that case. But this theorem is true for  F: X --> X'  where X is a prefoliation hence, even with AC, has no cleavage, and X' is an arbitrary category over S, i.e. has even less cleavages than X.
>
> In order not to make this mail too long I have not, but I should have, mentioned internalization where cleavages are even more problematic.
>
> Best to all,
> Jean
>
>
> Le 30 juil. 2014 ? 17:06, Thomas Streicher a ?crit :
>
>> Dear Jean,
>>
>> of course, you are right when emphasizing that one need choice for
>> classes to endow an "anonymous" fibration with a cleavage.
>> But that applies also to catgeories with say binary products. One
>> needs choice for classes in order to choose a product cone for every
>> pair of objects.
>> In many instances, however, categories come together with a choice of
>> products and fibrations come together with a choice of a cleavage.
>>
>> For example Set comes with a choice of a cleavage. Fibrations arising
>> from internal categories are even split. Many constructions on fibrations
>> allow one to choose a cleavage given cleavages for the arguments.
>> Do you know of any construction on fibrations which is not "cleavage
>> preserving" in this sense?
>>
>> Of course, one should not require cartesian functors to preserve
>> cleavages just as one should not require functors to preserve chosen
>> products.
>>
>> Thomas
>
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>



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Re: cleavages and choice

[Jean Bénabou]

Dear Eduardo,

Of course I fully agree with what you said. Let me just add a few remarks.
 
When Grothendieck defined fibrations it was for precise purposes, namely to axiomatize the notion of inverse image, and to use this axiomatization for descent (the titles of his talks are quite clear about that)

My view about fibrations is to try to get rid of sets, as much as possible, in category theory. This is what I tried to explain in my paper at the JSL.
Thus if some fibrations arising from set theory come equipped with more or less artificial cleavages, it does not impress me at all. 

The same process of elimination of sets, makes very sensitive, for fibrations but also for other domains, to the possibility of internalization.

Let me give an example of a totally different nature. In the paper where I introduced bicategories, their definition is in section 1.1 and takes 3 pages. But immediately after, it took me 10 pages to show in detail that they could be internalized, in any category with pullbacks. Let me point out that this paper was written in 1966 and published in 1967.

I think of course that ZFC has been a huge progress in the history of mathematics, but category theory has given us the possibility to explore new and fascinating countries. This goes for many domains of mathematics, and in particular.... for fibered categories.

Best to all,
Jean


Le 1 août 2014 à 18:47, Eduardo J. Dubuc a écrit :

> 
> Dear all,
> 
> I prefer fibrations over fibrations furnished with a cleavage (indexed categories) many times for reasons purely pragmatical mixed with an aesthetic philosophy.
> 
> Suppose you are dealing with fibrations where a canonical cleavage is present, suppose even that these cleavages come first and that the fibration is just a  conceptual context around them. Even in this case, faced to the need to produce a proof, if you succeed to find one without utilizing the cleavages, you will have something much nicer than the cleavage arguing. It will also give you a deeper understanding and a truthful light on the situation.
> 
> Suppose you do not care about foundations, axiom of choice, or things of that sort. You should still prefer fibrations. They are simpler, more to the point, and CERTAINLY AHEAD IN THE PROGRESS OF MATHEMATICS.
> 
> 
> NOTE: I wonder why so many people are so happy working with pull-backs and pull-back preserving functors (*) without even thinking in introducing a choice of pullbacks, and when it comes to fibrations, feel the need to introduce and work with cleavages.
> 
> (*) for example even when dealing with the category of sets (or categories whose objects have an underlying set), which are plenty of choices of pull-backs, for example, inverse image of a subset, the standard construction as a subset of the set of pairs, etc. We precisely teach in category theory courses that you should not work with any particular choice between the choices.
> 
> We all agree that it is neither necessary not good to choose a choice between all possible choices. This is precisely the progress that represents category theory thinking over set theory thinking. See for example, in the dawn of category theory, the considerations of Mac Lane concerning the fact that a quotient of a quotient of a group is not a quotient (as it is still now taught in algebra courses, category theory thinking has not arrived there yet).
> 
> You may say that the choice of a cleavage is at a different level than all this, but, essentially, deep down, for me it is the same. There is an old way of thinking (as Grothendieck said, SLN224, page 193) that hesitates in face of fibrations and prefer to work with a chosen cleavage.
> 
> 
> 

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Re: cleavages and choice

[Marco Grandis]

Dear Eduardo,

I agree with many things in your message, but I think you are taking your argument too far.
Talking of pullbacks you say:

> We precisely teach in category theory courses that you should not work with any
> particular choice between the choices.

I agree that it is better to avoid such a choice when possible. Yet you cannot define a bicategory of spans without assuming that such a choice has been made; in the same way as you cannot define the (good) monoidal structure of Ab without recurring to a choice of tensor products.
Such a situation, we all know, generally arises in non-strict bicategories (and monoidal categories, in particular).

Unless you want to redefine bicategories replacing the composition of arrows with an existence property. I still prefer working with a choice (eg of pullbacks) to such a complicated structure.

Best regards

Marco

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Re: cleavages and choice

[Eduardo J. Dubuc]

Dear Marco, supposing you are right (which is probably the case, I have
not examined closely the question of the monoidal structure of Ab or the
bicategory of spans), this does not contradicts my points. I never said
that choices should be banned, I said that they should not be used when
not necessary, because of plenty of different reasons.

best   e.d.


On 02/08/14 07:58, Marco Grandis wrote:
> Dear Eduardo,
>
> I agree with many things in your message, but I think you are taking
> your argument too far. Talking of pullbacks you say:
>
>> We precisely teach in category theory courses that you should not
>> work with any particular choice between the choices.
>
> I agree that it is better to avoid such a choice when possible. Yet
> you cannot define a bicategory of spans without assuming that such a
> choice has been made; in the same way as you cannot define the (good)
> monoidal structure of Ab without recurring to a choice of tensor
> products. Such a situation, we all know, generally arises in
> non-strict bicategories (and monoidal categories, in particular).
>
> Unless you want to redefine bicategories replacing the composition of
> arrows with an existence property. I still prefer working with a
> choice (eg of pullbacks) to such a complicated structure.
>
> Best regards
>
> Marco
>



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Re: cleavages and choice

[Thomas Streicher]

Dear Jean,

It has not been my intention to defend indexed categories against
fibered categories. I just wanted to say that it is sometimes
"convenient" to use cleavages.

First of all I would say that fibrations with cleavages are not the
same as indexed categories. Firstly, because the coherence conditions
for the maps chosen by the cleavage are determined by the functor and
do not have to be stated explicitly. Secondly, for fibrations with
cleavages one can show that they are closed under composition which
for indexed categories would be (close to) impossible.

Nevertheless, I agree with you that it is in general preferable to
formulate things in such a way that one avoids reference to cleavages
as far as possible. Sometimes, however, this makes things a bit more
complicated as I want to illustrate by the following example.

Chevalley's original formulation of his famous condition for internal
sums is much more convenient than the one usually found in the literature.
An analogue can be formulated for internal products (as in section 7
of my Notes on Fibrations you have mentioned). In Th.7.1 of loc.cit. I
have given a characterization of fibration having internal products which
avoids any reference to cleavages. This appears to me a bit clumsy and
there is a version using cleavages which essentially says that reindexing
functors have right adjoints and that their counits are preserved by
reindexing up to isomorphism. This latter version is useful when
checking that a given fibration has internal products as is necessary e.g.
when constructing models of type theory.

But in any case I think that conceptually it is preferable to define P
having internal products as P^op having internal sums. This formulation
is free from cleavages but for using it in concrete instances it is
sometimes useful to have equivalent formulations available which don't
abhor making reference to reindexing functors and thus to cleavages.

But this is a pragmatic issue and not a foundational issue. The same applies
to linear algebra. If it is convenient to refer to bases of vector spaces
I am not against doing so. But, of course, it would be stupid to require
all vector spaces to be endowed with bases.

For this reason I want to CORRECT the point of view of my previous mail.
One should not require all fibrations to be endowed with a cleavage. Rather
one should be open to accept some strong choice principles on the meta level
which allow one to assume the existence of cleavages whenever convenient.
But, definitely, one should give most definitions and constructions in a way
not referring to cleavages.

Thomas


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Re: cleavages and choice

[Paul Levy]

Related to Marco's point:

In order to define the category Set, we have to define Set(X,Y) for
all sets X and Y.  There are many isomorphic options, but to get a
category we have  to choose one.

Now replace "category" by "category with distinguished binary products
and exponentials.".  We shall then have to choose a particular
implementation of X x Y and X -> Y.  Why do some people find this
philosophically objectionable?  How is it worse than choosing
Set(X,Y)?  (Personally, I'd be inclined to make the same choice for X -
  > Y as for Hom(X,Y).)

Paul



On 2 Aug 2014, at 11:58, Marco Grandis wrote:

> Dear Eduardo,
>
> I agree with many things in your message, but I think you are taking
> your argument too far.
> Talking of pullbacks you say:
>
>> We precisely teach in category theory courses that you should not
>> work with any
>> particular choice between the choices.
>
> I agree that it is better to avoid such a choice when possible. Yet
> you cannot define a bicategory of spans without assuming that such a
> choice has been made; in the same way as you cannot define the
> (good) monoidal structure of Ab without recurring to a choice of
> tensor products.
> Such a situation, we all know, generally arises in non-strict
> bicategories (and monoidal categories, in particular).
>
> Unless you want to redefine bicategories replacing the composition
> of arrows with an existence property. I still prefer working with a
> choice (eg of pullbacks) to such a complicated structure.
>
> Best regards
>
> Marco
>

--
Paul Blain Levy
School of Computer Science, University of Birmingham
+44 121 414 4792
http://www.cs.bham.ac.uk/~pbl












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Re: cleavages and choice

[Toby Bartels]

Marco Grandis <grandis@dima.unige.it> wrote in part:

>Yet you cannot define a bicategory of spans
>without assuming that such a choice has been made;
>in the same way as you cannot define the (good) monoidal structure of Ab
>without recurring to a choice of tensor products.

A monoidal structure on a category C
is a functor C x C -> C, an object of C (aka a functor 1 -> C),
and various natural transformations satisfying some equations.
If by "functor" we mean an anafunctor, then no choice is needed.

Presumably you are thinking along these lines when you write

>Unless you want to redefine bicategories
>replacing the composition of arrows with an existence property.

My point is that anafunctors tell you automatically what to do.

Better yet, working in HoTT tells you automatically what to do.
All of this only looks complicated from a set-based perspective.


--Toby


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Re: cleavages and choice

[Eduardo J. Dubuc]

On 03/08/14 06:22, Thomas Streicher wrote:
> One should not require all fibrations to be endowed with a cleavage. Rather
> one should be open to accept some strong choice principles on the meta level
> which allow one to assume the existence of cleavages whenever convenient.
> But, definitely, one should give most definitions and constructions in a way
> not referring to cleavages.

I completely agree with this, it is my own feeling beautifully and
synthetically expressed. I only add:

   "... most definitions and constructions AND proofs in a way ..."

Also, that many times the use of cleavages may seem convenient, when in
fact it is not.

And I also put in the same bag the equivalences (which in general have
only one direction. Then one should proceed using the fully-faithfulness
and essentially surjectiveness whenever possible, and be free to assume
the existence of quasi-inverses only if it is necessary.


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Re: cleavages and choice

[Marco Grandis]

Dear Toby,

The theory of monoidal categories and bicategories is already written.
I may be interested in working with them, even though it is unpleasant to recur to choice for most (non-strict) structures of these kinds.
I have no interest in rewriting these theories in a different shape, using anafunctors or similar tools, or even study such variations if someone has given them.
Of course other people may prefer the way you are saying.
I just wanted to point out that there are many occasions where we are led to use choice - at least if we want to stay within 'classical' category theory, as expounded - say - in Mac Lane's text.
Regards,    Marco

 
On 03/ago/2014, at 18.30, Toby Bartels wrote:

> Marco Grandis <grandis@dima.unige.it> wrote in part:
> 
>> Yet you cannot define a bicategory of spans
>> without assuming that such a choice has been made;
>> in the same way as you cannot define the (good) monoidal structure of Ab
>> without recurring to a choice of tensor products.
> 
> A monoidal structure on a category C
> is a functor C x C -> C, an object of C (aka a functor 1 -> C),
> and various natural transformations satisfying some equations.
> If by "functor" we mean an anafunctor, then no choice is needed.
> 
> Presumably you are thinking along these lines when you write
> 
>> Unless you want to redefine bicategories
>> replacing the composition of arrows with an existence property.
> 
> My point is that anafunctors tell you automatically what to do.
> 
> Better yet, working in HoTT tells you automatically what to do.
> All of this only looks complicated from a set-based perspective.
> 
> 
> --Toby
> 


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