From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3374 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: Laws Date: Tue, 8 Aug 2006 09:38:02 +0100 (BST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241019267 8481 80.91.229.2 (29 Apr 2009 15:34:27 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:34:27 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue Aug 8 08:51:12 2006 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 08 Aug 2006 08:51:12 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1GAQ40-0007Vc-PS for categories-list@mta.ca; Tue, 08 Aug 2006 08:48:01 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 7 Original-Lines: 61 Xref: news.gmane.org gmane.science.mathematics.categories:3374 Archived-At: The following seems so obvious that I suspect it's not what Tom is really asking for; but it seems to me to be an answer to his question. A law in Tom's sense is just a parallel pair of arrows F(X) \rightrightarrows F(1) in the algebraic theory T under consideration (thinking of T as the dual of the category of finitely-generated free algebras). To get the theory of algebras satisfying a given set S of laws, you just need to construct the product-respecting congruence on T generated by S (i.e., the usual closure conditions for a congruence, plus the condition that f ~ f' and g ~ g' imply f x g ~ f' x g'), and factor out by it. Now any T-algebra A (in a category C, say) corresponds to a product- preserving functor F: T --> C; and the set of laws satisfied by A is just the (necessarily product-respecting) congruence generated by F, i.e. the set of parallel pairs in T having the same image under F. Is there anything more to it than that? Peter Johnstone ------------ On Mon, 7 Aug 2006, Tom Leinster wrote: > 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 > > > > >