Now for the question. Has there been any systematic study of what I have just defined as repletions? If not, are there in any case some papers I should consider?
I think the question as posed by David (relative to a particular category, in which some but not all diagrams of a particular shape may have colimits) is a very hard one. A lot is known about inferring the existence of particular types of colimits, in arbitrary categories, from the existence of other types: see the paper by Albert and Kelly "The closure of a class of colimits" in JPAA 51 (1988), and subsequent references of which Max will no doubt remind us. But when you work in a particular category, there are so many ways of "mutilating" the category by omitting particular objects which are required as the vertices of colimit cones, that I suspect there is almost nothing you can say in general. Incidentally, a similar comment applies to David's paper "Multilinearity of sketches" in TAC 3 (1997): I tried to make this point in my review (MR 98j:18006). Peter Johnstone