Postulated colimits and left exactness of Kan-extensions

If A is a small category and E a Grothendieck topos, the Kan extension LanF of a flat functor F : A → E along any functor A → D preserves whatever finite limits may exist in D; this is a well known fundamental result in topos theory. We shall present a metamathematical argument to derive out of this some other left exactness results, for Kan extensions with values in a (possibly large) site E. Loosely speaking, we prove that for any specific finite limit diagram (Di)i∈I inD, the Kan extension LanF preserves the limit diagram, provided the colimits in E used in the construction of the finitely many relevant Lan(Di)’s are what we call ‘postulated’ colimits. Both the notion of ‘flat’ F : A → E, and the notion of ‘postulated’ colimit in E are expressed in elementary terms in terms of the site structure (the covering notion) in E. If E is small with subcanonical topology, a colimit is postulated iff it is preserved by the Yoneda embedding of E into the topos Ẽ of sheaves on E. As a corollary, we shall conclude that if E satisfies the Giraud axioms for a Grothendieck topos, except possibly the existence of a set of generators (so E is an ∞-pretopos [2]), then any flat functor into E has left exact Kan extension (Corollary 3.3 below). We believe that the general method presented here is well suited to give partial left exactness results, by for instance allowing for coarse site structure on the recipent category E, so that there are relatively few flat functors into E. I am grateful to Bob Paré for bringing up the question of left Kan extensions with values in a ∞-pretopos, and for his impressive skepticism towards my original hand-waving change-of-universe arguments. I benefited much from several discussions we had on the subject. This work was carried out while we were both visiting Louvain-la-Neuve in May 1988. I want to express my gratitude to this university for its support and hospitality. Also, I want to thank Francis Borceux for bringing up, at the right moment,