From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3371 Path: news.gmane.org!not-for-mail From: "Tom Leinster" Newsgroups: gmane.science.mathematics.categories Subject: Laws Date: Mon, 7 Aug 2006 14:36:32 +0100 (BST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=iso-8859-15 Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1241019263 8470 80.91.229.2 (29 Apr 2009 15:34:23 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:34:23 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Aug 7 21:41:21 2006 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 07 Aug 2006 21:41:21 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1GAFal-0003tp-Rb for categories-list@mta.ca; Mon, 07 Aug 2006 21:37:07 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 4 Original-Lines: 37 Xref: news.gmane.org gmane.science.mathematics.categories:3371 Archived-At: Dear category theorists, Here's something that I don't understand. People sometimes talk about algebraic structures "satisfying laws". E.g. let's take groups. Being abelian is a law; it says that the equation xy = yx holds. A group G "satisfies no laws" if whenever X is a set and w, w' are distinct elements of the free group F(X) on X, there exists a homomorphism f: F(X) ---> G such that f(w) and f(w') are distinct. For example, an abelian group cannot satisfy no laws, since you could take X = {x, y}, w = xy, and w' = yx. There are various interesting examples of groups that satisfy no laws. To be rather concrete about it, you could define a "law satisfied by G" to be a triple (X, w, w') consisting of a set X and elements w, w' of F(X), such that every homomorphism F(X) ---> G sends w and w' to the same thing. A law is "trivial" if w = w'. Then "satisfies no laws" means "satisfies only trivial laws". You could then say: given a group G, consider the groups that satisfy all the laws satisfied by G. (E.g. if G is abelian then all such groups will be abelian.) This is going to be a new algebraic theory. What bothers me is that I feel there must be some categorical story I'm missing here. Everything above is very concrete; for instance, it's heavily set-based. What's known about all this? In particular, what's known about the process described in the previous paragraph, whereby any theory T and T-algebra G give rise to a new theory? Thanks, Tom