From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3379 Path: news.gmane.org!not-for-mail From: Rob Goldblatt Newsgroups: gmane.science.mathematics.categories Subject: Re: Laws Date: Wed, 9 Aug 2006 14:24:01 +1200 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019269 8506 80.91.229.2 (29 Apr 2009 15:34:29 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:34:29 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Aug 9 17:32:36 2006 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 09 Aug 2006 17:32:36 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1GAud2-00021R-PN for categories-list@mta.ca; Wed, 09 Aug 2006 17:26:12 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 12 Original-Lines: 122 Xref: news.gmane.org gmane.science.mathematics.categories:3379 Archived-At: One aspect of the categorisation of "equation" is the approach of Banaschewski and Herrlich. This replaces an equation as a pair (s,t) of terms that belong to some free algebra F by the least quotient (coequaliser) e: F -->> F/E identifying s and t. An algebra A satisfies equation (s,t) precisely when every homomorphism F --> A lifts across e to a homomorphism F/E --> A. Thus the notion of an equation becomes that of a regular epi e with free domain, and an object satisfies such an e when it is injective for e. To obtain a notion intrinsic to a given category, the free domains were replaced by domains that are regular-projective, i.e. projective for all regular epis, this being a property enjoyed by free algebras in categories of universal algebras. The approach has been dualised in the coalgebra literature, with a "coequation" being defined as a regular mono with regular-injective codomain, and a "covariety" as the class of coalgebras that are projective for some given class of coequations. Some (not all) relevant references are below. cheers, Rob @Article{ bana:subc76, author = "B. Banaschewski and H. Herrlich", title = "Subcategories Defined by Implications", journal = "Houston Journal of Mathematics", year = "1976", volume = "2", number = "2", pages = "149--171" } @Article{ adam:vari03, author = {Ji{\v{r}}{\'\i} Ad{\'a}mek and Hans-E. Porst}, title = "On Varieties and Covarieties in a Category", journal = "Mathematical Structures in Computer Science", year = "2003", volume = {13}, pages = "201-232" } @Techreport{ awod:coal00, author = "Steve Awodey and Jesse Hughes", title = "The Coalgebraic Dual of {B}irkhoff's Variety Theorem", institution = "Department of Philosophy, Carnegie Mellon University", year = "2000", number = "CMU-PHIL-109", note = "\url{http://phiwumbda.org/~jesse/papers/ index.html}" } On 8/08/2006, at 1:36 AM, 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 > > > >