From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6025 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: Tensor of monads Date: Mon, 2 Aug 2010 15:17:32 +0100 (BST) Message-ID: References: Reply-To: "Prof. Peter Johnstone" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: dough.gmane.org 1280788460 24102 80.91.229.12 (2 Aug 2010 22:34:20 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 2 Aug 2010 22:34:20 +0000 (UTC) Cc: Categories mailing list To: Richard Garner Original-X-From: majordomo@mlist.mta.ca Tue Aug 03 00:34:19 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Og3aH-00025H-Su for gsmc-categories@m.gmane.org; Tue, 03 Aug 2010 00:34:14 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:56350) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Og3ZA-0007mh-Vw; Mon, 02 Aug 2010 19:33:04 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Og3Z7-0002Zr-RJ for categories-list@mlist.mta.ca; Mon, 02 Aug 2010 19:33:01 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6025 Archived-At: Dear Richard, Your (*) is not an additional condition. Being a sheaf for both J and J' is equivalent to being a sheaf for their join (which I presume is what you mean by J n J'). For a proof, see A4.5.16 in the Elephant. Peter --------------------------- On Sun, 1 Aug 2010, Richard Garner wrote: > Further to my earlier question: > > -- Given idempotent monads S, T on a category C for which we can speak of >> the tensor of S and T, is it always the case that S * T is isomorphic to S + >> T? >> > > I think I'm now happy that the answer is "no". Consider, as in my previous > message, a presheaf category [D^op, Set]. Let S and T be the idempotent > monads corresponding to two > Grothendieck topologies J and J' on [D^op, Set]. Then S + T is the monad > whose algebras are presheaves which are simultaneously J-sheaves and > J'-sheaves. On the other hand, > S * T has as algebras those presheaves X which are both J-sheaves and > J'-sheaves, but which satisfy an additional axiom (*). This axiom may be > expressed most expediently when D has finite products; so let us assume that > now. The condition says: > > (*) Let f_i : U_i --> U be J-covering, and let g_k : V_k --> V be > J'-covering. Let ( x_ik \in X(U_i x V_k) ) be a compatible family for ( f_i > x g_k : U_i x V_j --> U x V ). Then the two natural ways of patching to an > element of X(U x V) agree. > > These two ways of patching are as follows. For the first, note that since ( > f_i x V_k | i \in I ) is J-covering for each k in K, we may patch to obtain > elements ( y_k \in X(U x V_k) | k \in K ). Then since ( U x g_k | k \in K ) > is J'-covering, we may patch these to obtain an element z \in X(U x V). For > the second way of patching, we proceed entirely analogously, but this time > going via a family ( y'_i \in X(U_i x V) | i \in I). > > Now (*) is a genuine extra condition which as far as I can see is not a > consequence of being both a J-sheaf and a J'-sheaf, so that S + T algebras > are not the same as S * T algebras. Note, however, that (*) _is_ a > consequence of being a (J n J')-sheaf, since ( f_i x g_j ) is covering in J > n J'. On the other hand, I'm not sure if (*) implies being a (J n J')-sheaf, > as I conjectured in my previous message; I don't have an Elephant to hand to > check. > > Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]