From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8117 Path: news.gmane.org!not-for-mail From: "George Janelidze" Newsgroups: gmane.science.mathematics.categories Subject: Re: Descent for fibred monads Date: Fri, 16 May 2014 09:22:27 +0200 Message-ID: References: Reply-To: "George Janelidze" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1400247546 14245 80.91.229.3 (16 May 2014 13:39:06 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 16 May 2014 13:39:06 +0000 (UTC) To: "Richard Garner" , "Categories list" Original-X-From: majordomo@mlist.mta.ca Fri May 16 15:39:01 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 1WlILk-00071t-Pn for gsmc-categories@m.gmane.org; Fri, 16 May 2014 15:39:00 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:58924) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1WlIL6-00050M-3V; Fri, 16 May 2014 10:38:20 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1WlIL4-0007ma-F1 for categories-list@mlist.mta.ca; Fri, 16 May 2014 10:38:18 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8117 Archived-At: Dear Richard, I would like to see your question formulated more precisely, and showing general (admissible) Galois theory example instead of the locally connected topos example. Some days ago you recommended Carboni-Janelidze-Kelly-Pare paper as one of references for factorization systems (thank you for that!), and now please look at Section 5 of that paper. Note that "admissible"="semi-left-exact" can be replaced with "fibration". Best regards, George Janelidze -------------------------------------------------- From: "Richard Garner" Sent: Thursday, May 15, 2014 1:15 PM To: "Categories list" Subject: categories: Descent for fibred monads > Dear categorists, > > Does the following variant of the Benabou-Roubaud/Beck monadic descent > theorem appear anywhere? > > Let p:E--->B be a fibration with sums and let T:E--->E be a fibred monad > over B. Let q: E^T ----> B be the induced fibration of T-algebras. Let > f: x--->y in B. Then to give T-algebra descent data for f---that is, a > diagram over the kernel-pair of f valued in E^T---is equally to give an > algebra for the composite monad > > E_x ----f_!----> E_y ----T_y---> E_y ---f^*----> E_x > > This doesn't seem to be an application of the usual monadic descent > theorem to q: E^T ---> B; that would identify T-algebra descent data for > f with algebras for a monad on (E^T)_x, not on E_x. > > For example, take E ----> S a connected topos with pi_0 -| Delta -| > Gamma. Let T be the monad for constant objects on E induced by the > fibred adjunction pi_0 -| Delta. Given f: U --->> 1 in E, to give > T-algebra descent data for f is to give a locally constant object split > by U. So such objects are equally the algebras for the monad > > E/U -----> E/U > (A--->U) |----> (Delta pi_0 A) x U ----> U > > In the same situation, take T to be the monad for free vector spaces E > ---pi_0---> S ---Fv---> S ---Delta---> E induced by the free vector > space monad Fv on S. Then T-algebra descent data over U --->> 1 is a > vector bundle split by U; so such objects are equally algebras for the > monad (A--->U) |----> (Delta Fv pi_0 A) x U ---> U > > Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]