I'm looking for a reference for the following lemma on composition of fibrations Let p:E->B and r:B->A be fibrations and D be a subcategory of A-> stable under pullback. If r has indexed products along morphisms in D and if p has indexed products along all cartesian maps above something in D then the composition rp has indexed products along D, too and the Beck-Chevalley condition holds. Martin Hofmann, Edinburgh ==============================================================================
what in the world is the Beck-Chevalley condition? ==============================================================================
Date: Wed, 2 Dec 92 08:13:34 GMT-0500 From: jds@rademacher.math.upenn.edu
what in the world is the Beck-Chevalley condition?
Let T : A^op --> Cat be a (pseudo) functor (such as obtained from a fibred category p : E --> A by taking T(a) to be the fibre of E over a in A). Suppose that A has pullbacks and that each T(f) : T(a) --> T(b) has an adjoint. Take a pullback square in A; apply T; now replace a pair of opposite sides in the square of functors by their adjoints. There is a canonical natural transformation in that square. We say T satisfies the Chevalley-Beck condition when this nat tran is invertible for all pullback squares. [See Benabou and Roubaud, Monades et descente, CR Acad Sc Paris 270 (1970) 96-98, and Lawvere, Equality in hyperdoctrines & comprehension schema as an adjoint functor, Proc. Symposia in Pure Math 17 (AMS 1970) 1-14.] The Chev-Beck cond is part of the requirement that T (as a category varying over A) should "have small coproducts" in the sense that every pointwise left kan extension into T, along a functor between small discrete variable categories, should exist. It is also required that each category T(a) should have small coproducts and each functor T(f) should preserve them. --Ross ==============================================================================
From: mxh@dcs.ed.ac.uk Date: Tue, 1 Dec 92 15:12:49 GMT
I'm looking for a reference for the following lemma on composition of fibrations [and the relation with indexed products].
There is a result which can be used in showing that every algebraic functor has an adjoint (Lawvere's thesis): if V is a cartesian closed category then the pointwise left kan extension k : A --> V of a finite product preserving functor f : B --> V, along any functor r : B --> A, is finite product preserving. [This can be tricky to prove, even for the expert. The only time I remember Saunders hesitating a bit during his marathon lecturing stint at Bowdoin College, Maine Summer 1969 was exactly over this point near the end of a lecture. He was trying to do it from the universal property of kan extensions. That evening Dubuc and I pointed out that pointwiseness was essential; by the next lecture Saunders had a proof which I'm sure he would have found without our comments!] Brian Day's Masters Thesis had a proof in the appendix. Borceux-Day have something that I have no time to look up in Bull Austral Math Soc (I think; help anyone?). Max Kelly and Stephen Lack have done something recently to appear in Appl Cat Structures Vol 1 (Univ Syd Math Report 92-29). But what the crumbs is Ross on about in relation to Martin Hofmann's question, you ask. Well, take V to be Cat, which is cartesian closed. Let p : E --> B be obtained from f : B --> Cat by the Grothendieck construction. Then r o p is the fibration constructed from k : A --> Cat. So, a reference? I'd say "its a variant of the above result on kan extns of prod pres funcs". It is also true that Cat is a topos wrt categories enriched over it, and then left kan extn of a (enriched) left exact functor into Cat is left exact. [See Kelly's paper in Cahiers on enriched locally presentable categories.] No time to give all the detail I'd like. --Ross ==============================================================================
Date: Wed, 2 Dec 92 08:13:34 GMT-0500 From: jds@rademacher.math.upenn.edu
what in the world is the Beck-Chevalley condition?
The Chevalley-Beck condition actually deals with bi-fibrations. Let p : E ---> B be a bi-fibration, and consider a pullback square in the base category B, say f A ----------> B | _| | | | h | | g | | V V C ----------> D k Probably the most intuitive case is the one of the codomain functor from the full subcategory of the arrow category Set^2 (where 2 denotes the two-element chain with 0 < 1) that has all monos as objects to Set. The fibres essentially are the power sets. It is a trivial exercise to show that it doesn't matter whether we first take the inverse image under h and then the direct image under f, or first the direct image under k and then the inverse image under g. The Chevalley-Beck condition generalizes this pleasant situation to arbitrary bi-fibrations. Consider a commutative square in E, say t X ----------> Y | | | | u | | v | | V V Z ----------> W w "over" the pullback above, such that u and v are cartesian (this corresponds to taking inverse images in the simple setting) and w is co-cartesian (this corresponds to to taking direct images). The Chevalley-Beck condition now states that t has to be co-cartesian as well. Other people on the net are better qualified to point out the significance of this condition. But I'd like to mention that even in the study of categorical closure operators this type of condition is the *right* one and arises naturally. This discovery was quite a pleasant surprize for me, after wrestling with unwieldy other conditions for a while. I hope this helps! -- J"urgen Koslowski | If I don't see you no more in this world Department of Mathematics | I meet you in the next world Kansas State University | and don't be late! koslowj@math.ksu.edu | Jimi Hendrix (Voodoo Chile) ==============================================================================
Maybe I should say that I do have a proof of the lemma in question, I was just wondering whether I should quote somebody. -- Martin Hofmann ==============================================================================
I'm looking for a reference for the following lemma on composition of fibrations
Let p:E->B and r:B->A be fibrations and D be a subcategory of A-> stable under pullback. If r has indexed products along morphisms in D and if p has indexed products along all cartesian maps above something in D then the composition rp has indexed products along D, too and the Beck-Chevalley condition holds.
Martin Hofmann, Edinburgh
I suppose you want to show 1) if r and p have enough indexed products, then rp has them; 2) if r and p satisfy the Beck-Chevalley (what-in-the-world) condition, then rp does - and then all this relativized with respect to a family of arrows D. ad 1) To construct the indexed products for rp, one needs NOT only enough indexed products in r and in p, but p must ALSO satisfy a weak form of the Beck-Chevalley-what-in-the-world condition. (I suppose the term "indexed products", as used in the query, does not include this condition - since it is singled out at the end.) The construction is in my thesis: Predicates and Fibrations, Rijksuniversiteit Utrecht 1990, prop. II.3.73 I think I had an example of two poset morphisms, which were fibrations, showing that the indexed products alone do not suffice. ad 2) A direct derivation of the Beck-Chevalley for the products in rp from this condition for the products in r and in p - seemed to me rather involved, because one had to chase those vertical-cartesian spans - the arrows of the opposite fibration (which is where the indexed products come about). It was simpler to forget the indexed products, and use my characterisation of the Beck-Chevalley-world condition for fibrations in general: Categorical interpolation: descent and the Beck-Chevalley-world condition without direct images, Springer Lecture Notes in Mathematics 1488 (The characterisation in this paper is not as simple as it could be, but I think it can be used.) Regards, Dusko Pavlovic P.S. Sorry. ==============================================================================
Date: Thu, 3 Dec 92 14:16:04 +1100 From: street@macadam.mpce.mq.edu.au
From: mxh@dcs.ed.ac.uk Date: Tue, 1 Dec 92 15:12:49 GMT
I'm looking for a reference for the following lemma on composition of fibrations [and the relation with indexed products].
There is a result which can be used in showing that every algebraic functor has an adjoint (Lawvere's thesis): if V is a cartesian closed category then the pointwise left kan extension k : A --> V of a finite product preserving functor f : B --> V, along any functor r : B --> A, is finite product preserving.
[This can be tricky to prove, even for the expert. The only time I remember Saunders hesitating a bit during his marathon lecturing stint at Bowdoin College, Maine Summer 1969 was exactly over this point near the end of a lecture. He was trying to do it from the universal property of kan extensions. That evening Dubuc and I pointed out that pointwiseness was essential; by the next lecture Saunders had a proof which I'm sure he would have found without our comments!] Brian Day's Masters Thesis had a proof in the appendix. Borceux-Day have something that I have no time to look up in Bull Austral Math Soc (I think; help anyone?). Max Kelly and Stephen Lack have done something recently to appear in Appl Cat Structures Vol 1 (Univ Syd Math Report 92-29).
A look at Lawvere's paper on "Metric spaces, Generalized Logic, and Closed Categories" might help as well. Bits and pieces of this may be found in "Basic Category Theory" (section 6.3 + 6.5) in: Abramsky, Gabbay, Maibaum (eds.) Handbook of Logic in computer Science, Vol.1, OUP 1992 Axel ------------------------------------------------------ Axel Poigne GMD I5 - SKS Schloss Birlinghoven Tel. + 2241 142440 fax. + 2241 142618 Postfach 1316 email poigne@gmd.de D - 5205 Sankt Augustin 1 ==============================================================================
so how did Chevally get in the act or is it not Claude? ==============================================================================
From: jds@rademacher.math.upenn.edu so how did Chevally get in the act or is it not Claude?
It is Claude: he used this condition in his seminar on descent in 1964. The fact that it is also a big thing in logic nowadays - it provides computer scientists with quantifiers - perhaps suggests that Lawvere (and some German philosophers) might be right about the unity of the worls. ==============================================================================
thanks you are the first to clarify this for me assume it is also Jon Beck? the unity of the world - ironic coming from a Bosnian Serb one can only hope that good work in Somalia will eventually come to bear on Bosnia jim ==============================================================================
From: jds@rademacher.math.upenn.edu (Jim Stasheff)
the unity of the world - ironic coming from a Bosnian Serb
I am afraid it's worse than that. In Greece, the irony used to be the essence of the tragedy: the distance which allows public to watch the bloody clashes of the antagonists, resulting in death and the unity. In Bosnia, "the tragedy repeats itself" as the reality (not "as a farce", as the quotation would go) --- stripped of the irony. Instead of heroic katarsis, there is this fascist "clensing" of the helpless. And there is no deus ex machina. I am sorry, I am obvously saying what doesn't belong here, talking to suppress thoughts perhaps, but there is no irony. Dusko ==============================================================================
participants (6)
-
jds＠rademacher.math.upenn.edu -
koslowj＠math.ksu.edu -
mxh＠dcs.ed.ac.uk -
pavlovic＠triples.Math.McGill.CA -
poigne＠gmd.de -
street＠macadam.mpce.mq.edu.au