On Wed, Jul 13, 2011 at 9:10 PM, Toby Bartels <categories@tobybartels.name> wrote:
the axiom of collection, which implies that we can find some *set* of objects containing *at least one* limit for every finite diagram in the original small subcategory; and then we can iterate countably many times to obtain a small category which contains at least one limit for any finite diagram therein.
The axiom of collection guarantees only *some* appropriate set of objects, so you need to choose one. To iterate this countably many times, you might need dependent choice.
That's a good point. However, I think we can get around it as follows. We can make finitely many choices without any axiom of choice. Thus, for any natural number n, by applying collection n times, we can find *some* n^th iterate of the "construction". (Formally, we prove this by induction on n.) Applying the axiom of collection again over the natural numbers, we obtain a set which contains at least one n^th iterate of the "construction" for every natural number n. Taking the union of this set, we should obtain a set of objects whose corresponding full subcategory contains at least one limit of every finite diagram therein. Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]