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
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