From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6026 Path: news.gmane.org!not-for-mail From: Paul Levy Newsgroups: gmane.science.mathematics.categories Subject: Re: Tensor of monads Date: Mon, 2 Aug 2010 20:55:01 +0100 Message-ID: References: Reply-To: Paul Levy NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v936) Content-Type: text/plain; charset=ISO-8859-1; format=flowed; delsp=yes Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1280788505 24234 80.91.229.12 (2 Aug 2010 22:35:05 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 2 Aug 2010 22:35:05 +0000 (UTC) Cc: Categories mailing list To: Richard Garner Original-X-From: majordomo@mlist.mta.ca Tue Aug 03 00:35:03 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 1Og3b0-0002Mb-Kr for gsmc-categories@m.gmane.org; Tue, 03 Aug 2010 00:34:58 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:56372) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Og3Zx-0007pd-GS; Mon, 02 Aug 2010 19:33:53 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Og3Zs-0002cB-Mo for categories-list@mlist.mta.ca; Mon, 02 Aug 2010 19:33:48 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6026 Archived-At: On 1 Aug 2010, at 01:31, Richard Garner wrote: > I must admit to feeling slightly confused by both Peter's and Andr=E9's > examples. In both cases, the monads considered arise on a category =20 > other > than the category of sets; and it is not clear to me what is meant by > forming the tensor product of two such monads. Here is a suggestion; I don't know how it relates to yours. Let S and T be monads on a category C. An "S,T-algebra" is a C-object X together with an S-algebra structure =20= theta and T-algebra structure phi. An S,T-algebra morphism is a C-=20 morphism that is homomorphic in both components. Let D be the =20 category of S,T-algebras and homomorphisms, and U : D --> C the =20 forgetful functor. Then U creates U-split coequalizers. If it has a =20= left adjoint, we call the monad the "sum" of S and T. I think the sum of S and T, if it exists, has to be a coproduct in the =20= category of monads, but haven't checked the details. Next suppose that C is cartesian, and S and T are strong. Now D will =20= be a locally C-indexed (by this I mean [C^op,Set]-enriched) category. =20= A morphism from (X,theta,phi) to (X',theta',phi') over Z is a C-=20 morphism from Z x X to X' that's homomorphic in its second argument, =20 with respect to both structures. If U has a (locally C-indexed) left =20= adjoint, we get the "sum" of strong monads. Again, I think it's a =20 coproduct in the category of strong monads. Next suppose C is cartesian closed and S and T are strong. For an S,T-algebra (X,theta,phi), the following are equivalent: (1) for all C-objects Y and Z, the two C-morphisms from SY x TZ x =20 X^(YxZ) to X are equal (2) for every C-object Y, the two C-morphisms from SY x T(X^Y) to X =20 are equal (2') for every C-object Z, the two C-morphisms from TZ x S(X^Z) to X =20 are equal. When these hold, we say that (X,theta,phi) "commutes". (I'd like to =20 express this without quantification over objects, but I can't see how.) We thus obtain a full (locally C-indexed) subcategory D' of D =20 consisting of the commuting S,T-algebras and homomorphisms, and U' : =20 D' --> C the restriction of U. Then U' creates U'-split =20 coequalizers. If it has a left adjoint, we call the induced monad the =20= "tensor" of S and T. Now a cocone of strong monads S -----> M <----- T is said to "commute" when for all C-objects X and Y the two C-=20 morphisms from SX x TY to M(X x Y) are equal. I think a tensor of S and T will always give an initial commuting =20 cocone, but haven't checked the details. Paul -- Paul Blain Levy School of Computer Science, University of Birmingham +44 (0)121 414 4792 http://www.cs.bham.ac.uk/~pbl [For admin and other information see: http://www.mta.ca/~cat-dist/ ]