Steve Lack – 15 April 1998
I discussed briefly the well-known construction of the free completion P(C) of a category C under a class P of colimits (meaning, of course, a class of weights). If C is small, this is formed as the closure of C in [C^op,Set] under P-colimits.
I then looked at interated cocompletions Q(P(C)), where P and Q are different classes, and saw that this is still a full subcategory of [C^op,Set] provided that P-limits commute with Q-colimits. If Q is taken to be the class of all colimits commuting with P-limits, then for some but not all choices of P, it is the case that Q(P(C)) is all of [C^op,Set]. This condition on P holds if and only if for every P-continuous functor f:A-->Set, the left Kan extension Lan_Y(f):[A^op,Set]-->Set is also P-continuous.
Next I looked at some particular classes P. If P is any of the following classes: finite limits k-small limits finite products connected limits terminal object finite connected limits then the condition of the previous paragraph holds, but if P is the class of pullbacks, it does not.
The class of colimits (meaning of course the class of weights) commuting with pullbacks is the same as the class of colimits commuting with finite connected limits. They can be formed as the iterated cocompletion Q(P(C)) where P is the class of filtered colimits and Q is the class of coproducts; or alternatively they can be formed as the itereated cocompletion P(R(C)) where P is again the class of filtered colimits, but R is the class of finite coproducts. As a corollary of the equivalence of Q(P(C)) and P(R(C)), we deduce that Fam(K) is finitely accessible if K is so. More generally, Fam(K) is k-accessible if K is k-accessible; this strengthens a result of Makkai and Paré which proves that Fam(K) is accessible if K is so.