Dear Francois, I don't know how relevant this is to your thinking, but I thought I'd mention the result is constructively false. For an example, let C be the poset with two elements b <= t. The only idempotents are the identities on b and t, so of course they split. Let p be a truth value, and let I = {b} u {t | p}, an ideal (hence directed) in C. If this has a colimit, then it must be either b or t. The colimit is b iff not p (so I = {b}), and it follows that if the colimit is t we have not not p. Hence existence of the colimit gives (not p or not not p) for every p, which is not intuitionistically valid. In fact one way to regard the set of truth values (i.e. the subobject classifier) is as the ideal completion of C. Regards, Steve. On 21 Apr 2008, at 20:04, lamarche wrote:
Fellow category theorists,
I'm looking for a ref for the following result:
Let C be a finite category. Then TFAE
-- C has colimits for all small filtered (well, directed is probably better) diagrams.
-- Idempotents split in C.
This doesn't seem to be in Makkai-Paré or Adamek-Rosicky, but surely somebody must have observed this. Actually the result is a bit more general since idempotents split in *any* category that has filtered colimits.
Thanks in advance,
François Lamarche