From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6018 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Tensor of monads Date: Sun, 1 Aug 2010 19:13:55 +1000 Message-ID: References: Reply-To: Richard Garner NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1280752185 5985 80.91.229.12 (2 Aug 2010 12:29:45 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 2 Aug 2010 12:29:45 +0000 (UTC) To: Categories mailing list Original-X-From: majordomo@mlist.mta.ca Mon Aug 02 14:29:45 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 1Ofu9I-0007RC-9K for gsmc-categories@m.gmane.org; Mon, 02 Aug 2010 14:29:44 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:51647) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Ofu8D-0008Mj-S1; Mon, 02 Aug 2010 09:28:37 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Ofu8B-0006iP-MA for categories-list@mlist.mta.ca; Mon, 02 Aug 2010 09:28:35 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6018 Archived-At: 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/ ]