From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8114 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Descent for fibred monads Date: Thu, 15 May 2014 21:15:44 +1000 Message-ID: Reply-To: Richard Garner NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1400206403 13015 80.91.229.3 (16 May 2014 02:13:23 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 16 May 2014 02:13:23 +0000 (UTC) To: Categories list Original-X-From: majordomo@mlist.mta.ca Fri May 16 04:13:18 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 1Wl7e3-0004zc-0R for gsmc-categories@m.gmane.org; Fri, 16 May 2014 04:13:11 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:58800) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1Wl7d4-0002kY-JA; Thu, 15 May 2014 23:12:10 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Wl7d4-0005XM-OI for categories-list@mlist.mta.ca; Thu, 15 May 2014 23:12:10 -0300 X-Mailer: MessagingEngine.com Webmail Interface - ajax-d19652da Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8114 Archived-At: 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/ ]