From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6036 Path: news.gmane.org!not-for-mail From: Gordon Plotkin Newsgroups: gmane.science.mathematics.categories Subject: Re: Tensor of monads Date: Sun, 8 Aug 2010 20:24:27 +0100 Message-ID: References: Reply-To: Gordon Plotkin NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1281354883 26174 80.91.229.12 (9 Aug 2010 11:54:43 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 9 Aug 2010 11:54:43 +0000 (UTC) Cc: categories@mta.ca To: Sergey Goncharov Original-X-From: majordomo@mlist.mta.ca Mon Aug 09 13:54:41 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 1OiQwB-0002vI-Lz for gsmc-categories@m.gmane.org; Mon, 09 Aug 2010 13:54:40 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:55258) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OiQvD-00064V-VI; Mon, 09 Aug 2010 08:53:39 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OiQv5-0003FQ-Et for categories-list@mlist.mta.ca; Mon, 09 Aug 2010 08:53:31 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6036 Archived-At: Dear Sergey, In the paper by Hyland, Levy, Power and myself, Combining algebraic effects with continuations, there is a proof that the tensor of the continuations monad R^(R^-) (|R| >=3D 2) with itself, or, indeed with any monad T with a constant (i.e. such that T(0) is not empty), is the trivial monad. It is not hard to see this directly, via large Lawvere theories. The large Lawvere theory L of the continuations monad has: L(X,Y) =3D Set(R^X,R^Y) and so the constants L(0,1) correspond to maps 1 --> R. Further, using two distinct constants, any two operations R^X --> R^Y can be coded up into one operation R^(X +1) --> R^Y and then recovered via the two constants. Given maps of large Lawvere theories L_T ---> M <----L such that the images of any two operations in L_T and L commute, as L_T has a constant all (the images of) constants in L are identified, as usual, but then so are all images of any two operations R^X --> R^Y (which will, for example, include all pairs of projections) and so M is trivial. A more general theorem is also proved in the paper which has as a consequence that the tensor of any monad with rank with the continuations monad exists. On Thu, Aug 5, 2010 at 9:06 PM, Sergey Goncharov wrote: > Thank you Peter and Andr=E9 > and all the participants of the discussion. It is indeed very helpful. > > Richard Garner wrote: >> >> On the other hand, Andr=E9's example raises a question which I find quit= e >> interesting. Andr=E9 describes two reflective subcategories of the order= ed >> class of ordinal numbers, and then says that, their intersection being >> empty, the tensor of the corresponding idempotent monads cannot exist. I >> would be inclined to say that this shows that the coproduct of these >> monads >> does not exist > > I guess it applies both to the tensor and to the sum as well as to any ot= her > case where we need to form a span of monad morphism: S -> R <- T and whic= h > precisely can not be formed in this case. > > It looks like there are two counterexamples, both of which are based on t= he > construction of tricky underlying categories. But what about existence of > the tensor over Sets? I guess this is still open. =A0I tried to think abo= ut > the tensor product of a continuation monad with itself as a possible > counterexample, without any success though. Usually, continuation monad > performs well when it comes to constructing counterexamples but it is > difficult to see what the tensor product of it with itself =A0is supposed= to > look like. > > Thanks again, > Sergey. > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]