On 6/17/2009 3:45 PM, Steve Lack wrote:
Hmm. Not sure if you mean you're allowing any full subcategory of [J^op,Set]; if so then you should drop the requirement that J-->C be fully faithful. By "category of presheaves on J" I had in mind retaining J as part of it.
Am I missing something? I was thinking that followed from density of J in C.
No. The category Setf of finite sets has a fully faithful dense inclusion in to the (presheaf) category Set of all sets, but Set is not [Setf^op,Set].
Oops, right, I was mixing up cocomplete and cocompletion-of. (Actually I don't think in terms of either, I find it easier to think of [J^op,Set] as the maximal dense extension of J up to equivalence, in the sense that all dense extensions of J are full subcategories of it.) Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]