From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8157 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Composition of Fibrations and Quantification Date: Tue, 10 Jun 2014 15:42:36 +1000 Message-ID: References: Reply-To: Richard Garner 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 1402410047 14042 80.91.229.3 (10 Jun 2014 14:20:47 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 10 Jun 2014 14:20:47 +0000 (UTC) To: Steve Vickers , Categories Original-X-From: majordomo@mlist.mta.ca Tue Jun 10 16:20:41 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 1WuMul-0007vV-Ec for gsmc-categories@m.gmane.org; Tue, 10 Jun 2014 16:20:39 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:35282) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1WuMtj-0007e2-DE; Tue, 10 Jun 2014 11:19:35 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1WuMth-0002tY-EG for categories-list@mlist.mta.ca; Tue, 10 Jun 2014 11:19:33 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8157 Archived-At: Dear Neil, Steve, Steve: what you say about fibrations and opfibrations is completely correct. So if E ---> D is a bifibration, and D ---> C is a bifibration, then the composite E ---> C is a bifibration. It is moreover easy to check that if the individual parts of this satisfy Beck-Chevalley, then so too will the composite. Thus the composite of fibrations with sums is again a fibration with sums. For products, though, I think the situation is a bit more complex. I could not see how to derive the result by a simple duality. From scribbling on the back of an envelope, it does appear to be true that: - if E -p--> D and D ---q---> C are fibrations with products, then so too is E ---qp---> C The argument I have is basically the following. Given e in E with pe =3D d and qd =3D c, and some f: c ---> c' in C, we wish to describe the Pi of e along f; we'll write this as f_*(e). Well first we form f_*(d) in D, its pullback g : f^*f_*(d) ----> f_*(d) along f, and the counit k: f^*f_*(d) ---> d in the fibre over c. Using these data in D, we now form in E the pullback k^*(e) of e along k and then the Pi along g, so yielding e' =3D g_*k^*(e). Now pe' =3D f_*(d) and q(f_*(d)) =3D c' and it's now not so hard to check that e' is in fact the Pi of e along f, as desired. For the checking in the last step, I seemed to need a few times the Beck-Chevalley condition for products. So I doubt one could get away without that. I have not tried to verify whether the Pi's of the composite so defined themselves satisfy the Beck-Chevalley condition, but I would be amazed if this were not the case. Finally, one might ask the following question. Suppose that E ----> D and D ----> C are fibrations with products and sums, and that in each case the sums and products satisfy the distributivity axiom ("type-theoretic axiom of choice"). Does the composite fibration, which by the above again has sums and products, also satisfy the distributivity axiom? I do not know the answer to this, but I rather suspect so. It would be a largeish diagram chase. It would be nice to know if there was a more abstract reason why this is true. Richard On Sun, Jun 8, 2014, at 11:58 PM, Steve Vickers wrote: > Some thoughts: >=20 > * The result about composition of fibrations holds in any 2-category with > comma objects and 2-pullbacks, not just Cat. (Think of the Chevalley > criterion for fibrations.) >=20 > * By duality on 2-cells it thus also applies to opfibrations, and hence > to bifibrations. >=20 > * It is bifibration structure that gives you the left adjoints you ask > for. >=20 > * For the right adjoints, look at the dual 2-category, where your > fibrations become bifibrations. >=20 > Hence it seems to me that your conjectures are all true, and even > generalize widely. >=20 > Steve. >=20 >> On 6 Jun 2014, at 10:47, Neil Ghani wrote: >>=20 >> Dear All >>=20 >> We know that if p and q are fibrations, then their composition p.q is a= fibration. >>=20 >> But what about quantification =E2=80=A6 that is if reindexing along eve= ry morphism 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? >>=20 >> Thanks for any thoughts >> Neil [For admin and other information see: http://www.mta.ca/~cat-dist/ ]