From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8160 Path: news.gmane.org!not-for-mail From: Claudio Hermida Newsgroups: gmane.science.mathematics.categories Subject: Re: Composition of Fibrations and Quantification Date: Fri, 13 Jun 2014 00:28:48 +0100 Message-ID: References: Reply-To: Claudio Hermida NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1402686139 4529 80.91.229.3 (13 Jun 2014 19:02:19 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 13 Jun 2014 19:02:19 +0000 (UTC) Cc: Categories To: Neil Ghani Original-X-From: majordomo@mlist.mta.ca Fri Jun 13 21:02:14 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 1WvWjs-0000En-79 for gsmc-categories@m.gmane.org; Fri, 13 Jun 2014 21:02:12 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:35973) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1WvWiz-00032s-Sp; Fri, 13 Jun 2014 16:01:17 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1WvWiw-0007Qi-IJ for categories-list@mlist.mta.ca; Fri, 13 Jun 2014 16:01:14 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8160 Archived-At: Dear All We know that if p and q are fibrations, then their composition p.q is a fibration. But what about quantification =E2=80=A6 that is if reindexing along every m= orphism has a right/left adjoint in p and q, then does reindexing along every morphism in p.q have a right/left adjoint? Under some circumstances? Thanks for any thoughts Neil Dear Neil, Instead of quantification, I will refer to the structure you ask about as products (in the case of right adjoints) and coproducts (in the case of left adjoints) for a fibration, which is a more standard terminology in fibred category theory. Notice that as Thomas pointed out, one considers such adjoints subject to Beck-Chevalley conditions, i.e., all products/coproducts are required to be given pointwise. With regard to products in a composite fibration, it probably helps if we start by recalling a simple result about ordinary products and fibrations. *PROP* *[1]*: Let p: E -> B be a fibration, and B admit (finite) products. The following are equivalent i) E admits (finite) products and p preserves them ii) p has fibred (finite) products Here ii) means every fibre has (finite) products, preserved by reindexing. To formulate the equivalent result for a composite of fibrations, the key is to regard such a composite gadget as a fibration between fibrations. Namely, given fibrations F: E -> D and b: D -> B, consider the composite fibration t =3D bF: E -> B. Now, F is a fibred functor F: t -> b (over B), that is, a morphism in the 2-category Fib/B Benabou proved the following equivalent: a) F is a fibration in Fib/B b) F is a fibration in Cat Item i) means that every fibre F_I:E_I -> D_I is a fibration and cartesian morphsims of such fibrations are stable under reindexing. Hence, the given fibration F can be profitably viewed as a fibration between the fibrations t and b. Now we can reproduce the first result: *PROP*: Assume the base fibration b has products. The following are equivalent i) t has products and F preserves them ii) F has fibred products Just as in the Cat case, ii) means that every fibre fibration F_I: E_I -> D_I has products, and these later are stable under reindexing. It is what results from spelling out =E2=80=98fibration with products=E2=80=99 in the = 2-category Fib/B. All the relevant definitions can be found in Jacobs=E2=80=99 book, I= think. One can specialize the products relative to certain classes of morphisms, e.g., projections from cartesian products. Playing with the various dualities on bases and fibres, one gets similar results for coproducts. With regards to proofs, I will just note that ALL the results above, namely, existence of products, fibration in Cat =3D fibration in Fib/B, and the fact that the composite of 2 fibrations is again a fibration are direct consequences of the lifting/factorization of adjunctions in the 2-fibration cod:Fib -> Cat *[2]* References: *[1]* *Gray, John W.* Fibred and cofibred categories. 1966 *Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) *pp. 21--83*Springer, New York= * *[2] **Hermida, Claudio*. Some properties of *Fib* as a fibred 2-category. = *J. Pure Appl. Algebra* *134 *(1999), no. 1, 83--109. Claudio [For admin and other information see: http://www.mta.ca/~cat-dist/ ]