Re: Fundamental Theorem of Category Theory?

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

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


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