Notions of Flatness Relative to a Grothendieck Topology

Completions of (small) categories under certain kinds of colimits and exactness conditions have been studied extensively in the literature. When the category that we complete is not left exact but has some weaker kind of limit for finite diagrams, the universal property of the completion is usually stated with respect to functors that enjoy a property reminiscent of flatness. In this fashion notions like that of a left covering or a multilimit merging functor have appeared in the literature. We show here that such notions coincide with flatness when the latter is interpreted relative to (the internal logic of) a site structure associated to the target category. We exploit this in order to show that the left Kan extensions of such functors, along the inclusion of their domain into its completion, are left exact. This gives in a very economical and uniform manner the universal property of such completions. Our result relies heavily on some unpublished work of A. Kock from 1989. We further apply this to give a pretopos completion process for small categories having a weak finite limit property.