From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4321 Path: news.gmane.org!not-for-mail From: "Stephen Lack" Newsgroups: gmane.science.mathematics.categories Subject: Re: The Category of Semimodules over Semirings Date: Mon, 17 Mar 2008 13:25:27 +1100 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019870 12748 80.91.229.2 (29 Apr 2009 15:44:30 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:44:30 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Mon Mar 17 09:32:38 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 17 Mar 2008 09:32:38 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JbEP2-0002de-EV for categories-list@mta.ca; Mon, 17 Mar 2008 09:25:20 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 85 Original-Lines: 50 Xref: news.gmane.org gmane.science.mathematics.categories:4321 Archived-At: As Fred says, the semimodules over a given semiring are=20 determined by operations and equations, and so are complete and cocomplete. In terms of exactness properties they are also=20 (i) locally finitely presentable (so that finite limits commute with filtered colimits, and=20 (ii) Barr-exact (so that there is an equivalence between quotients and congruences) If we restrict to the cancellative case, we still have completeness and cocompleteness and (i), but (ii) fails. =20 Steve Lack. > -----Original Message----- > From: cat-dist@mta.ca [mailto:cat-dist@mta.ca] On Behalf Of=20 > Fred E.J. Linton > Sent: Monday, March 17, 2008 4:43 AM > To: categories@mta.ca > Subject: categories: Re: The Category of Semimodules over Semirings >=20 > On Sun, 16 Mar 2008 09:26:35 AM EDT Jawad Abuhlail=20 > wrote, in part, on the Subject: The=20 > Category of Semimodules over Semirings, >=20 > > ... The category of semimodules had products, equalizers=20 > and products=20 > > (however not necessarily coequalizers). >=20 > I must be missing something here. Don't the (say, left-)=20 > semimodules (over a given semiring) constitute an=20 > equationally definable class of algebras? That is, aren't=20 > they determined entirely by operations and equations? >=20 > If they DO, that is, if they ARE, then the category of them=20 > all (together with their homomorphisms) must, like all such=20 > "varietal categories," have all (small) limits and colimits,=20 > and, in particular, all coequalizers. >=20 > Alas, I have little else to offer. Cheers, and Happy St. Paddy's Day, >=20 > -- Fred >=20 >=20 >=20 >=20 >=20