From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5534 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: Re: Examples for the Yoneda lemma Date: Fri, 15 Jan 2010 14:57:30 +1100 Message-ID: References: Reply-To: Steve Lack NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1263598460 26446 80.91.229.12 (15 Jan 2010 23:34:20 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 15 Jan 2010 23:34:20 +0000 (UTC) To: Hans-Peter Stricker , categories Original-X-From: categories@mta.ca Sat Jan 16 00:34:12 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 1NVvgC-0002qH-DK for gsmc-categories@m.gmane.org; Sat, 16 Jan 2010 00:34:12 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NVvCM-0002xq-U9 for categories-list@mta.ca; Fri, 15 Jan 2010 19:03:23 -0400 Thread-Topic: categories: Examples for the Yoneda lemma Thread-Index: AcqVltvf6cC7lsBZU0Kk4dmS+AKYjw== In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5534 Archived-At: Dear Hans-Peter, In the case you mention, the category of graphs is a presheaf category (the category of all functors C^op-->Set, for a small category C. The graphs V and E you mention are precisely the representable functors, and the morphisms V to E are the morphisms between these. This is a general phenomenon: for any presheaf category [C^op,Set], you can recover an object X when you know all the morphisms to X from the representables, as well as how these morphisms behave when composed with morphisms between representables. The key property here is that the representables form a dense full subcategory of [C^op,Set], and there is a further generalization to that setting. For another concrete example, consider the category Ab of abelian groups. The elements of a group A can be identified with the morphisms Z-->A. Pairs of elements can be identified with morphisms Z^2-->A, and the group operation can then be recovered using the diagonal map Z-->Z^2. The other operations can be recovered similarly (you'd also want to use the trivial group in order to recover the unit element). If you also want to check the associative law, you'd want to use the object Z^3 as well. The connection between the previous two paragraphs is that the full subcategory of Ab consisting of Z, Z^2, Z^3, and 1 is dense. (Actually, you could just use Z and Z^2 for this.) A similar analysis can be done for the models of any Lawvere theory in place of Abelian groups. Steve Lack. On 15/01/10 11:24 AM, "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/ ]