From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5537 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 13:01:52 +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 1263598937 27638 80.91.229.12 (15 Jan 2010 23:42:17 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 15 Jan 2010 23:42:17 +0000 (UTC) To: Hans-Peter Stricker , categories@mta.ca Original-X-From: categories@mta.ca Sat Jan 16 00:42:10 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 1NVvnt-0005Ve-69 for gsmc-categories@m.gmane.org; Sat, 16 Jan 2010 00:42:09 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NVvXA-00044v-Hp for categories-list@mta.ca; Fri, 15 Jan 2010 19:24:52 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5537 Archived-At: Here, it's important to distinguish graph and category. A (small) category is a graph in some sense, but not all graphs are necessarily categories. For instance, a category must have a composition operation E x E -> E that is closed on edges. As far as the poset-embedding example goes, Todd Trimble has an excellent article on the topic: http://topologicalmusings.wordpress.com/2008/04/02/toward-stone-duality-posets-and-meets/ Maybe this will clarify. - Aleks On Fri, Jan 15, 2010 at 12:50 PM, Hans-Peter Stricker wrote: > Hello Aleks, > > I am not quite what to think of the poset of unlabeled graphs without > isolated vertices with the relation of embeddability: I have the feeling, > that such a graph is NOT completely determined by its set of in-arrows (see > http://epublius.de/Fragment_of_the_category_of_unlabeled_graphs_without_isolated_vertices.pdf > to see what I mean, e.g. vertices 3 and 4 or vertices 7,8,9). > > Do I miss something? > > Best > Hans-Peter > > ----- Original Message ----- From: "Aleks Kissinger" > To: "Hans-Peter Stricker" > Cc: > Sent: Friday, January 15, 2010 12:07 PM > Subject: Re: categories: Examples for the Yoneda lemma > > >> 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/ ] >>> > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]