The goal of this lecture, motivated by a question of Lawvere, is to introduce a notion of tight completion, which consists in an external enrichment of the original category, unlike completions achieved through cohesive variation.
Instances of tight completions are both the Cauchy completion of a V-category for V a complete closed category with a faithful functor into Sets (Lawvere 1973), and the Grothendieck completion, or stack completion of the Kaorubi envelope (Bunge 1979), of an S-category for S a Grothendieck topos - the latter of central importance in non-abelian cohomology of a topos (Giraud 1972, Grothendieck 1973).
Generalizing from S to V, I obtain the (surprising) result that the Cauchy and the Grothendieck completions of a V-category C are in fact equivalent. To this end, I exhibit first an equivalence between the category of (V-adjoint) V-distributors (Benabou 1973) with target C and the category of (V-essential) generalized V-functors (Lawvere 1966) with target C.
The Axiom of Choice ("Sets") is of central importance in these investigations, as it is inherent to the notion of a stack ("within Geometry"). I give here several instances of this phenomenon and, in particular, clarify and complete some statements made in the literature on Cauchy completions (Borceux and Dejean 1986).