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-pose... Maybe this will clarify. - Aleks On Fri, Jan 15, 2010 at 12:50 PM, Hans-Peter Stricker <stricker@epublius.de> 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_isol... 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" <aleks0@gmail.com> To: "Hans-Peter Stricker" <stricker@epublius.de> Cc: <categories@mta.ca> 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 <stricker@epublius.de> 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
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