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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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.
... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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))
... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
participants (4)
-
Eduardo J. Dubuc -
Jean Bénabou -
Steve Vickers -
Thomas Streicher