From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8120 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 20:53:52 +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 1400369089 20017 80.91.229.3 (17 May 2014 23:24:49 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 17 May 2014 23:24:49 +0000 (UTC) To: "Richard Garner" , "Categories list" Original-X-From: majordomo@mlist.mta.ca Sun May 18 01:24:48 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 1WlnyB-0001Tc-BD for gsmc-categories@m.gmane.org; Sun, 18 May 2014 01:24:47 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:59145) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1Wlnxg-0001Tl-Tr; Sat, 17 May 2014 20:24:16 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Wlnxg-0004YY-KS for categories-list@mlist.mta.ca; Sat, 17 May 2014 20:24:16 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8120 Archived-At: Dear Richard, I am sorry, but, unless I completely misunderstood what you are saying, what you call "(2)" is simply wrong. Moreover, this can be seen in the 'very first" example of Galois theory. For, take: (a) C to be the category of G-sets, where G is any fixed non-trivial group; (b) X to be the category of sets; (c) I -| H to be what you called "pi_0 -| Delta" in your first message (that is, for A in C, I(A) is the set of orbits of A, while for S in X, H(S) is the set S equipped with the trivial action of G); (d) B = 1, the one-element G-set; (e) E = G, considered as a G-set, on which G acts via its multiplication. Then C / E is equivalent to the category of sets, and in particular each of its objects is a coproduct of copies of its terminal object G=G; and let us calculate your monad, which is sufficient to do for G=G: (g) Your C / E --Sum_p--> C / B sends G=G to G-->1; (h) Then I^B sends G-->1 to 1=1, the terminal object of X / I(B) = X / 1; (i) H^B and p^* preserves the terminal object; (j) that is, your monad sends G=G to G=G, and so it is the identity monad. But the right monad is the free G-set monad (if we identify C / E with the category of sets). Please either confirm or explain what have I misunderstood in your message. George -------------------------------------------------- From: "Richard Garner" Sent: Friday, May 16, 2014 10:29 AM To: "George Janelidze" ; "Categories list" Subject: categories: Re: Descent for fibred monads > 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 > > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]