From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8217 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Jean_B=E9nabou?= Newsgroups: gmane.science.mathematics.categories Subject: Re: Composition of Fibrations Date: Sun, 20 Jul 2014 18:18:57 +0200 Message-ID: Reply-To: =?iso-8859-1?Q?Jean_B=E9nabou?= NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Apple Message framework v1283) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1406053977 15504 80.91.229.3 (22 Jul 2014 18:32:57 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 22 Jul 2014 18:32:57 +0000 (UTC) To: Steve Vickers , Categories Original-X-From: majordomo@mlist.mta.ca Tue Jul 22 20:32:51 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1X9ero-0001t9-Ur for gsmc-categories@m.gmane.org; Tue, 22 Jul 2014 20:32:49 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:47634) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1X9eqs-0002Ho-Lq; Tue, 22 Jul 2014 15:31:50 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1X9eqs-0002ut-0b for categories-list@mlist.mta.ca; Tue, 22 Jul 2014 15:31:50 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8217 Archived-At: 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.=20 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.=20 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.=20 (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.=20 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.=20 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.=20 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.=20 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..=20 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.=20 But it has neither cleavages, cocleavages, nor reindexings.=20 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.=20 However, even if S =3D 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.=20 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.=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]