Re: Fundamental Theorem of Category Theory?

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

Steve Lack wrote:
> Your proposed characterization is actually a characterization of full
> subcategories of [J^op,Set] containing the representables.

Right, that's what I meant by "*a* category of presheaves on J" (as
opposed to *the* category of all presheaves on J), the point of my
analogy with Archimedean fields (as opposed to the field of all reals).

> To get the whole
> presheaf category you should add that C is cocomplete,

Right, just as to get all of the reals one should say that the
Archimedean field is complete.  For situations where one doesn't need
the whole thing it is convenient to be able to characterize the
categorical counterpart of an Archimedean field, with J in place of Q,
as any full, faithful and dense extension of J.  Density serves to keep
the extension inside [J^op,Set], just as it keeps Archimedean fields
inside R.

> and that homming out
> of objects in J is cocontinuous (i.e. C(j,-) is cocontinuous for all j in
> J).

Am I missing something?  I was thinking that followed from density of J
in C.

Vaughan


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