Re: Examples for the Yoneda lemma

[Aleks Kissinger <aleks0@gmail.com>]

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
<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_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" <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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>>>
>
>


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