From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5533 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Examples for the Yoneda lemma Date: Thu, 14 Jan 2010 20:45:50 -0800 Message-ID: References: Reply-To: Vaughan Pratt NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1263598450 26420 80.91.229.12 (15 Jan 2010 23:34:10 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 15 Jan 2010 23:34:10 +0000 (UTC) To: categories list Original-X-From: categories@mta.ca Sat Jan 16 00:34:03 2010 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 4.50) id 1NVvg2-0002mQ-5l for gsmc-categories@m.gmane.org; Sat, 16 Jan 2010 00:34:02 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NVvEp-00035C-ND for categories-list@mta.ca; Fri, 15 Jan 2010 19:05:55 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5533 Archived-At: Lots of examples in http://boole.stanford.edu/pub/yon.pdf . It's an 18-page paper, yet already by page 2 there are six examples, none of them the usual graph example. In coming to grips with those examples it is *very* helpful to realize that every algebraic theory including the well-known ones having at least one constant or constant operation has a unary subtheory obtained by fixing all but one argument of every nonzeroary operation in the clone (which will be just projection if and only if all nonzeroary operations are projections). And while you don't need it on page 2, further on it is helpful to realize that for every algebraic theory, its models form a full subcategory of a presheaf category. In order to reach a broader audience the paper was written for an algebraic audience. If you're more familiar with category language than algebraic language a little adaptation will be needed. Let me know if translating back into category theory gives any trouble, this would be helpful feedback to have. Vaughan Pratt Hans-Peter Stricker wrote: > Hello, > > I am looking for (simple) instructive examples for the Yoneda lemma, > showing > how to get the "inner" structure of an object from its morphisms. I've been > told how to get a graph G from its morphisms (from the one-vertex-graph > V to > G and the one-edge-graph E to G and the morphisms from V to E) and > appreciated this example a lot. Are there others equally simple and > enlightening? > > What I wonder is which morphisms are definitely needed. In the graph > example > it's the morphisms from V -> G, E -> G and V -> E? Can this be abstracted > and generalized? > > Many thanks in advance > > Hans-Stricker [For admin and other information see: http://www.mta.ca/~cat-dist/ ]