From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6017 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 10:31:47 +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 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1280752153 5893 80.91.229.12 (2 Aug 2010 12:29:13 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 2 Aug 2010 12:29:13 +0000 (UTC) To: Categories mailing list Original-X-From: majordomo@mlist.mta.ca Mon Aug 02 14:29:10 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 1Ofu8k-0007GW-Be for gsmc-categories@m.gmane.org; Mon, 02 Aug 2010 14:29:10 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:51634) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Ofu7H-0008J7-Uf; Mon, 02 Aug 2010 09:27:39 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Ofu7F-0006gv-8o for categories-list@mlist.mta.ca; Mon, 02 Aug 2010 09:27:37 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6017 Archived-At: 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 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. In the case of finitary monads on Set, the meaning is clear, since a finitary monad corresponds to = a Lawvere theory, a Lawvere theory is a special kind of small category with finite products, and we know how to form the tensor product of two categories with finite products (essentially because the doctrine of finite products is a pseudo-commutative one). This extends without problem to monads with rank on Set; and even to monads without rank on Set, so long as one recognises that the correlate notion of Lawvere theory will be a large one, so that the tensor product may not always exist. In the case of base categories other than Set, one would have to use a generalisation of the notion of Lawvere theory: such has been given by Nishizawa-Power (see also Lack-Power) but they require that the base category be locally presentable, which is not the case in the examples of Andr=E9 and Peter. My own take on what Peter's example is doing is the following. Given any finitary monad L on Sets, there is a corresponding Lawvere theory T, and so for any category C, we can consider the category Mod(T,C) of T-models in C: it's the category of finite-product preserving functors T -> C. There is a forgetful functor Mod(T,C)-->C given by evaluation at 1, and this will be monadic so long as it has a left adjoint; in which case we induce a monad L= ' on C. In particular, letting L be the finitary monad on Set whose algebras are sup-semilattices-without-a-unit, letting L * L be its tensor with itself, and letting C be the category of finite sets of odd cardinality, then Peter's example seems to show that: -- The induced monad L' on C exists but the induced monad (L * L)' does not. On the other hand, Andr=E9's example raises a question which I find quite interesting. Andr=E9 describes two reflective subcategories of the ordered 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, but this leads on to my question, which is: -- 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? Here is a test case. Power's 2000 paper "Enriched Lawvere theories" shows that, for a symmetric monoidal closed V which is lfp as a closed category, finitary V-monads are equivalent to enriched Lawvere theories, which are certain small V-categories with finite cotensors. Using the tensor of such V-categories, we can define a tensor of such monads. Consider in particular when V is [D^op, Set] for some small D, and let S and T be two idempotent strong (=3DV-enriched) monads on V. Now it should be possible to calculate = S + T and S * T in terms of the corresponding Lawvere theories and to see if they coincide. I haven't tried this yet but it should be an interesting exercise. (The obvious thing to try first is to take S and T corresponding to sheaf subtoposes of [D^op, Set]. Then these monads, being product-preserving, are commutative, and so it's natural to think that S * = T should be the monad corresponding to the intersection of these subtoposes. = S + T, on the other hand, I'm inclined to think may be something bigger, and not necessarily cartesian.) Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]