From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8118 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Descent for fibred monads Date: Fri, 16 May 2014 18:29:22 +1000 Message-ID: References: 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 1400247587 14848 80.91.229.3 (16 May 2014 13:39:47 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 16 May 2014 13:39:47 +0000 (UTC) To: George Janelidze , Categories list Original-X-From: majordomo@mlist.mta.ca Fri May 16 15:39:42 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 1WlIMO-0000HL-Ln for gsmc-categories@m.gmane.org; Fri, 16 May 2014 15:39:40 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:58930) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1WlILl-00057J-J8; Fri, 16 May 2014 10:39:01 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1WlILk-0007nO-1P for categories-list@mlist.mta.ca; Fri, 16 May 2014 10:39:00 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8118 Archived-At: Dear George, Thanks for your message. I should say that the locally connected topos example was just intended to be a sample application of the modified monadic descent theorem quoted at the start of my message. But as you point out, one could also apply it in the setting of categorical Galois theory that you refer to. In the terminology of [CJKP] this would say something like: Let I -| H : X ----> C be an admissible reflection, and p: E --> B an effective descent map in C. Then Spl(E,p) is isomorphic to the category of algebras for the monad C / E --Sum_p--> C / B --H^B.I^B--> C / B --p^*--> C / E Which decomposes into the two statements: (1) Let I -| H : X ----> C be an admissible reflection, and p: E --> B an effective descent map in C. Then Spl(E,p) is isomorphic to the category of M-on-objects discrete fibrations over the kernel-pair \bar B of p --- i.e. C^{\bar B} /\ M / {\bar B} in the terminology of [CJKP] -- which is (part of) the Theorem on p.26 of ibid.; and (2) Let I -| H : X ----> C be an admissible reflection, and p: E --> B _any_ map in C. Then C^{\bar B} /\ M / {\bar B} is isomorphic to the category of algebras for the monad C / E --Sum_p--> C / B --H^B.I^B--> C / B --p^*--> C / E --- and it is really this (2) which I am interested in. Does this come up in the categorical Galois theory literature? The other example I attempted in my original message, but botched rather badly, involving vector bundles, was an attempt to give some application of this modified descent theorem to a fibred monad which is not a fibred reflection. The categorical Galois theory example is compelling because one has a fibred monad whose algebras do not descend along effective descent morphisms (although, of course, the underlying objects do). The point is that the algebras for lots of fibred monads DO descend along effective descent morphisms, e.g., any fibred monad on a topos E ----> S induced by a finitary algebraic theory in S. So I guess a subsidiary question is whether there are any compelling examples of non-idempotent fibred monads whose algebras do not descend. Richard On Fri, May 16, 2014, at 05:22 PM, George Janelidze wrote: > 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/ ] > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]