Dear Vaughan, Your proposed characterization is actually a characterization of full subcategories of [J^op,Set] containing the representables. To get the whole presheaf category you should add that C is cocomplete, and that homming out of objects in J is cocontinuous (i.e. C(j,-) is cocontinuous for all j in J). Steve. On 16/06/09 7:58 AM, "Vaughan Pratt" <pratt@cs.stanford.edu> wrote:
Apropos of the Yoneda Lemma, is there some reason why it is usually stated on its own rather than as one direction of a characterization of categories of presheaves on J? Unless I've overlooked or misunderstood something it seems to me that the Yoneda Lemma should state that C is a category of presheaves on J if and only if there exists a full, faithful, and dense functor from J to C.
This should generalize the characterization of an Archimedean field as any dense extension of the rationals.
Vaughan Pratt
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