Re: Examples for the Yoneda lemma

[Steve Lack <s.lack@uws.edu.au>]

Dear Hans-Peter,

In the case you mention, the category of graphs is a presheaf category (the
category of all functors C^op-->Set, for a small category C. The graphs V
and E you mention are precisely the representable functors, and the
morphisms V to E are the morphisms between these.

This is a general phenomenon: for any presheaf category [C^op,Set], you can
recover an object X when you know all the morphisms to X from the
representables, as well as how these morphisms behave when composed with
morphisms between representables.

The key property here is that the representables form a dense full
subcategory of [C^op,Set], and there is a further generalization to that
setting.

For another concrete example, consider the category Ab of abelian groups.
The elements of a group A can be identified with the morphisms Z-->A. Pairs
of elements can be identified with morphisms Z^2-->A, and the group
operation can then be recovered using the diagonal map Z-->Z^2. The other
operations can be recovered similarly (you'd also want to use the trivial
group in order to recover the unit element). If you also want to check the
associative law, you'd want to use the object Z^3 as well.

The connection between the previous two paragraphs is that the full
subcategory of Ab consisting of Z, Z^2, Z^3, and 1 is dense. (Actually, you
could just use Z and Z^2 for this.)

A similar analysis can be done for the models of any Lawvere theory in place
of Abelian groups.

Steve Lack.





On 15/01/10 11:24 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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