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 ==============================================================================