Re: no fundamental theorems please

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

Vaughan Pratt wrote:
> ...  The Yoneda Lemma usually states that J embeds (meaning
> fully) in [J^op,Set], ...

Dear Vaughan,

The usual statement is significantly stronger than that (see e.g. Mac
Lane, Mac Lane and Moerdijk, or Wikipedia). It says that, for
contravariant functors F: C -> Set, the elements of FX are in bijection
with transformations to F from the representable functor for X. Your
statement can be deduced by considering the particular case where F too
is representable.

(To put it another way, the representable presheaf for X is freely
generated - as presheaf - by a single element (the identity morphism) at
X. This then allows you to calculate the left adjoint of the forgetful
functor from presheaves over C to ob(C)-indexed families of sets.)

There can be no doubt that this strong Yoneda Lemma is vitally important
when calculating with presheaves - for example, it shows immediately how
to calculate exponentials and powerobjects. If F and G are two
presheaves, then the exponential G^F is calculated by

   G^F(X) = nt(Y(X), G^F)    (by Yoneda's Lemma)
          = nt(Y(X) x F, G)  (by definition of exponential)

I don't think you can get it and its useful consequences from your
weaker statement, even if you start strengthening yours in the way you
suggest by supplying converses.

Another closely related and important result, though not known as
Yoneda's Lemma as far as I know, is that the presheaf category over C is
a free cocompletion of C, and the Yoneda embedding is the injection of
generators.

(By the way, I agree that category theory doesn't have to have a
Fundamental Theorem. I haven't see any compelling reason to appoint one.)

Regards,

Steve.


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