From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8316 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Is the category of group actions LCCC? Date: Fri, 05 Sep 2014 11:17:00 +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 1409921152 18417 80.91.229.3 (5 Sep 2014 12:45:52 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 5 Sep 2014 12:45:52 +0000 (UTC) Cc: categories@mta.ca To: Ross Street , Timothy Revell Original-X-From: majordomo@mlist.mta.ca Fri Sep 05 14:45:46 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.127]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1XPsta-0000Xa-CD for gsmc-categories@m.gmane.org; Fri, 05 Sep 2014 14:45:42 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:42525) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XPstC-0004Ii-11; Fri, 05 Sep 2014 09:45:18 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XPst9-0005Qs-Gu for categories-list@mlist.mta.ca; Fri, 05 Sep 2014 09:45:15 -0300 X-Mailer: MessagingEngine.com Webmail Interface - ajax-a219acad In-Reply-To: <1409879112.2347407.163846569.68720436@webmail.messagingengine.com> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8316 Archived-At: A small correction. As well as: Let p: E ----> B be a fibration. If B is cartesian closed, each fibre is cartesian closed with exponents stable under pullback, and Pi's exist along product projections (and satisfy BCC), then E is cartesian closed. the following stronger statement is true (weakening the stability required of the exponents): Let p: E ----> B be a fibration. If B is cartesian closed, each fibre is cartesian closed with exponents stable under pullback along product projections, and Pi's exist along product projections (and satisfy BCC), then E is cartesian closed. In order to capture Ross' example, this stronger form is needed, since [f,1]:[B,Set] ---> [A,Set] does not in general preserve exponentials, while [pi_2,1]:[B,Set] ---> [A*B,Set] does so. Richard >=20 > The product of (a, phi) with (b, psi) in E is of course (a x b, > pi_1^*(phi) x pi_2^*(psi)) with pi_1 : a <--- a x b ----> b : pi_2 the > product projections in B. >=20 > The internal hom [(b, psi), (c, gamma)] is Pi_{pi_1} [pi_2^*(psi), > ev^*(theta)], where pi_1 : [b,c] <---- [b,c] x b ----> b : pi_2 and=20 > ev: [b,c] x b ----> c in B. >=20 > This in particular applies to Cat//=E2=80=99Set=E2=80=99 as in Ross' mess= age, seen as a > fibration over Cat with reindexing along f:A--->B given by > [f,1]:[B,Set]--->[A,Set]. This fibration has right adjoints to > pullbacks, but they don't satisfy BCC; however, right adjoints to > pullback along product projections are given just by (conical) limit > functors, and these do satisfy BCC. So the preceding construction > applies (and a bit of fiddling about shows that this does indeed agree > with Ross' prescription). >=20 > As for local cartesian closure: if B is lccc, each fibre is lccc with > fibrewise Pi's stable under pullback, and E--->B has all products, then > it seems that each slice fibration p/A: E/A--->B/pA will satisfy the > conditions in the second paragraph, whence E is also lccc. >=20 > Richard >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]