From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5876 Path: news.gmane.org!not-for-mail From: James Borger Newsgroups: gmane.science.mathematics.categories Subject: filtered monads? [resent] Date: Thu, 27 May 2010 13:06:14 +1000 Message-ID: Reply-To: James Borger NNTP-Posting-Host: lo.gmane.org Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1275152561 29304 80.91.229.12 (29 May 2010 17:02:41 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 29 May 2010 17:02:41 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Sat May 29 19:02:40 2010 connect(): No such file or directory Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OIPQk-0004fb-LG for gsmc-categories@m.gmane.org; Sat, 29 May 2010 19:02:38 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OIPAn-0007Gc-Fc for categories-list@mta.ca; Sat, 29 May 2010 13:46:09 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5876 Archived-At: Dear category theorists, Does the concept of "filtered monad" exist in the literature? Here are = two basic models of what I have in mind. 1. Let C be the category of sets, let F:C->C be the set underlying the = free monoid on S, and let F_n(S) be the subset of F(S) consisting of = words of length at most n. Then the monad structure map F o F-->F = restricts to maps F_m o F_n-->F_{mn}, and F_1 is the identity functor. 2. Let C be the category of R-modules (R a given ring), F(M) is the = tensor product R[x] \otimes_R M, and F_n(M) is the sub-R-module M + xM + = ... + x^n M of F(M). Then the monad structure map F o F --> F restricts = to a map F_m o F_n --> F_{m+n}, and F_0 is the identity functor. So, in the first example, I'd say that the monad F is filtered by the = ordered monoid of non-negiative integers under multiplication, and in = the second example, it's filtered by that under addition. There seems to be a pretty obvious way of formalizing this, and since = many monads in practice come with such a structure, I'd guess that this = concept is in the literature, but I didn't find anything on the internet = or in the textbooks on my shelf. But perhaps that's because it's not = called a "filtered monad" or because it's a special case of a general = concept with a completely different name. So, does this concept exist in = the literature? I'm writing something about a particular monad with a a = filtered structure, and after I define it, I'd like to have the sentence = "In the language of [5], the functors F_n endow F with a filtered monad = structure." Yours, James Borger ps I'm not at the moment a subscriber to the mailing list, so please cc = to me any responses to the list. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]