From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8122 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Descent for fibred monads Date: Sat, 17 May 2014 17:16:33 +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 1400369198 21284 80.91.229.3 (17 May 2014 23:26:38 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 17 May 2014 23:26:38 +0000 (UTC) To: George Janelidze , Categories list Original-X-From: majordomo@mlist.mta.ca Sun May 18 01:26:33 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 1Wlnzr-0008Tq-QT for gsmc-categories@m.gmane.org; Sun, 18 May 2014 01:26:31 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:59157) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1WlnzI-0001no-MO; Sat, 17 May 2014 20:25:56 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1WlnzI-0004bW-7E for categories-list@mlist.mta.ca; Sat, 17 May 2014 20:25:56 -0300 In-Reply-To: <6322FED48A6B4BA486625A8E350B1BD5@ACERi3> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8122 Archived-At: Ah! You are quite correct. I was hasty in saying that the Galois theory situation is an example of the result I am interested in. The reason it does not work is that the reflection HI does not induce a fibred monad on C. The semi-left-exactness ensures the simple formula for the reflection: A--->B goes to the pullback of HIA ----> HIB along B---->IHB. What it does not ensure is that pullback commutes with reflection---which would be to ask that HI be left exact. This deficiency also applies to the example I started with, of a locally connected topos E---->S. The "fibred monad" Delta pi_0 is only fibred over S, whereas I need it to be fibred over E. So in fact it seems that a correct example is given by a topos with totally connected components --- meaning that the left adjoint pi_0 of Delta preserves pullbacks. In this case, then, the analogue of (2) does hold. Richard On Sat, May 17, 2014, at 04:53 AM, George Janelidze wrote: > 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 > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]