From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8223 Path: news.gmane.org!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: Composition of Fibrations Date: Tue, 22 Jul 2014 15:55:44 +0100 Message-ID: References: <3E52EFB7-7955-47B1-9B00-9F6F6152BBC1@cs.bham.ac.uk> <32AB43B0-58DA-4375-A4FD-6C84F4E527EA@wanadoo.fr> <6EFFC44F-E933-412B-89F2-C33B598D78B0@cs.bham.ac.uk> <9747FDFD-FF71-4ACE-8DD3-538462A1B283@wanadoo.fr> Reply-To: Steve Vickers NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (1.0) Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1406054502 22040 80.91.229.3 (22 Jul 2014 18:41:42 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 22 Jul 2014 18:41:42 +0000 (UTC) Cc: Categories To: =?utf-8?Q?Jean_B=C3=A9nabou?= Original-X-From: majordomo@mlist.mta.ca Tue Jul 22 20:41:36 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 1X9f0J-0000kI-RX for gsmc-categories@m.gmane.org; Tue, 22 Jul 2014 20:41:36 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:47676) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1X9f01-0003ZO-FQ; Tue, 22 Jul 2014 15:41:17 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1X9f00-0003BA-Oj for categories-list@mlist.mta.ca; Tue, 22 Jul 2014 15:41:16 -0300 In-Reply-To: <9747FDFD-FF71-4ACE-8DD3-538462A1B283@wanadoo.fr> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8223 Archived-At: [Ross: Any comments?] Dear Jean, I wasn't properly aware of those issues around choosing iso or equality, so i= t's lucky I got into this discussion. My intuition of what you are saying is= that with iso, roughly speaking, the reindexing is only pseudofunctorial. Fo= r example, if you pull back along the diagonal B -> =CE=A6B, to get the rein= dexing along identities, then you get an endofunctor of each fibre that is i= somorphic to the identity. Am I on the right lines? Is this all written down= somewhere? After your messages I noticed that Street has a remark after his proposition= , whose significance I overlooked: "Compare the above proposition with Gray [2] p.56; so we have related the de= finition of 0-fibration here with the definition of opfibration in [2] when K= =3D Cat. Notice that the unit of the adjunction l -| p~ for Gray is not jus= t an isomorphism but an identity. It is worth pointing out the reason for th= is since we will need the observation in the next paper. A 0-fibration will b= e called normal when there is a normalized pseudo L-algebra structure on it.= In Cat every 0-fibration is normal, but in other categories this need not b= e the case. In the proof of the Chevalley criterion, if =CE=B6 is an identit= y then so is =CE=B7. So, for a normal 0-fibration, p~ : =CE=A6E -> p/B has a left adjoint with unit an identity." Notes: 1. Gray [2] =3D "Fibred and cofibred categories", La Jolla. 2. I don't know which paper Street means by "the next paper". 3. In a pseudo L-algebra E, with structure morphism c: LE -> E, =CE=B6 denot= es the isomorphism from Id_E to unit composed with c. E is normalized if =CE= =B6 is equality. 4. =CE=B7 is the unit of the adjunction. But that seems to claim that in Cat it doesn't matter whether you use iso or= equality in the Chevalley condition. Does that accord with your understandi= ng? Regards, Steve. > On 22 Jul 2014, at 05:24, Jean B=C3=A9nabou wrot= e: >=20 > Dear Steve, >=20 > At least one ambiguity is solved. Chevalley gave as criterium (for opfibra= tions) that the arrow=20 > p~ : =CE=A6E -> p/B in your mail has a left adjoint with unit the identit= y. > When the 2-category is Cat this condition is satisfied iff p is an opfibr= ation which has an opcleavage. The choice of the adjoint defines the opcleav= age. >=20 > Let us for the sake of precision call Street criterium the existence of a= left adjoint with unit an iso, and Street opfibrations (in Cat) the functor= s which satisfy this condition.=20 > They need not be opfibrations in the sense of Grothendieck which is almost= unanimously adopted. It is unfortunate to have given them the name of (op)f= ibrations, not only because of the ambiguity as we have seen, but because th= e fibers are meaningless, in particular the fibers over two isomorphic objec= ts of the base B need not be isomorphic.=20 >=20 > I'm almost sure that Neil Ghani, Richard Garner, Claudio Hermida and Thoma= s Streicher meant Grothendieck fibrations, and the genuine Chevalley conditi= on in your answer, as I did. >=20 > Regards, >=20 > Jean >=20 >=20 >=20 >> Dear Jean, >>=20 >> Street's result is as follows. The arrow p: E -> B is a 0-fibration over B= if and only if the arrow >> p~ : =CE=A6E -> p/B >> corresponding to the 2-cell >>=20 >> =CE=A6E --pd1--> B >> | || >> d0 || >> | p=CE=BB =3D> || >> v || >> E --p------> B >>=20 >> has a left adjoint with unit an isomorphism. >>=20 >> Here =CE=A6E =3D E/E and p/B are comma objects, d0 and d1 are projections= , and =CE=BB is the canonical 2-cell in a comma square (in this case for =CE= =A6E). 0-fibration is opfibration. >>=20 >> Regards, >=20 >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]