Re: Examples for the Yoneda lemma

[Vaughan Pratt <pratt@cs.stanford.edu>]

Lots of examples in http://boole.stanford.edu/pub/yon.pdf .  It's an
18-page paper, yet already by page 2 there are six examples, none of
them the usual graph example.

In coming to grips with those examples it is *very* helpful to realize
that every algebraic theory including the well-known ones having at
least one constant or constant operation has a unary subtheory obtained
by fixing all but one argument of every nonzeroary operation in the
clone (which will be just projection if and only if all nonzeroary
operations are projections).  And while you don't need it on page 2,
further on it is helpful to realize that for every algebraic theory, its
models form a full subcategory of a presheaf category.

In order to reach a broader audience the paper was written for an
algebraic audience.  If you're more familiar with category language than
algebraic language a little adaptation will be needed.  Let me know if
translating back into category theory gives any trouble, this would be
helpful feedback to have.

Vaughan Pratt

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


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