On the existence and exactness of the associated sheaf functor

AbstractLet C be a category with inverse limits. A category xis called an A-topos if there is a site (ϱ, τ), i.e. a small category ϱ together with a Grothendieck topology τ such that xis equivalent to the category Shτ[C0. A] of τ-sheaves on C with values in C. If xis an C-topos, then so is Shτ'[ϱ0, x] for any site (ϱ', τ'). It is shown that if for every site (ϱ,τ) the associated sheaf functor from presheaves to τ-sheaves with values in A exists (and preserves finite inverse limits), then the same holds if Ais replaced by any A-topos x. Roughly speaking, the main result is that for a site (ϱ,τ) the associated sheaf functor [ϱ0, A] → Shτ [ϱ0, A] exists and preserves finite inverse limits, provided A has filtered direct limits which commute with finite inverse limits, e.g. if A is a Grothendieck category or a category of sheaves with values in a locally finitely presentable category [8. 7.1]. Analogous results hold in the additive case