From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5531 Path: news.gmane.org!not-for-mail From: Aleks Kissinger Newsgroups: gmane.science.mathematics.categories Subject: Re: Examples for the Yoneda lemma Date: Fri, 15 Jan 2010 11:07:53 +0000 Message-ID: References: Reply-To: Aleks Kissinger NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: ger.gmane.org 1263598257 25973 80.91.229.12 (15 Jan 2010 23:30:57 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 15 Jan 2010 23:30:57 +0000 (UTC) To: Hans-Peter Stricker , categories@mta.ca Original-X-From: categories@mta.ca Sat Jan 16 00:30:50 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 1NVvcv-0001a6-Tj for gsmc-categories@m.gmane.org; Sat, 16 Jan 2010 00:30:50 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NVvHq-0003Dl-Ew for categories-list@mta.ca; Fri, 15 Jan 2010 19:09:02 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5531 Archived-At: The simplest example I can think of is posets. If you represent a poset as a category (i.e. a category with at most one arrow from A->B such that A->B and B->A implies A=B), then an object A is completely determined by the set of arrows going in to it. In this context, the Yoneda embedding is the familiar result that any poset P embeds fully and faithfully in the powerset of P, ordered by subset inclusion. Aleks On Fri, Jan 15, 2010 at 12:24 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/ ]