Tightly Bounded Completions

By a ‘completion’ on a 2-category K we mean here an idempotent pseudomonad on K . We are particularly interested in pseudomonads that arise from KZdoctrines. Motivated by a question of Lawvere, we compare the Cauchy completion [23], defined in the setting of V-Cat for V a symmetric monoidal closed category, with the Grothendieck completion [7], defined in the setting of S-Indexed Cat for S a topos. To this end we introduce a unified setting (‘indexed enriched category theory’) in which to formulate and study certain properties of KZ-doctrines. We find that, whereas all of the KZ-doctrines that are relevant to this discussion (Karoubi, Cauchy, Stack, Grothendieck) may be regarded as ‘bounded’, only the Cauchy and the Grothendieck completions are ‘tightly bounded’ – two notions that we introduce and study in this paper. Tightly bounded KZ-doctrines are shown to be idempotent. We also show, in a different approach to answering the motivating question, that the Cauchy completion (defined using ‘distributors’ [2]) and the Grothendieck completion (defined using ‘generalized functors’ [21]) are actually equivalent constructions.