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 KZ-doctrines. Motivated by a question of Lawvere, we compare the Cauchy completion, defined in the setting of V-Cat for V a symmetric monoidal closed category, with the Grothendieck completion, 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') and the Grothendieck completion (defined using `generalized functors') are actually equivalent constructions.
Keywords: 2-categories, KZ-doctrines, completions, enriched category theory, indexed categories, distributors, generalized functors, Karoubi envelope, Stack completion, Cauchy completion, Grothendieck completion
2010 MSC: 18A25, 18A40, 18B25, 18C15, 18D10, 18D15, 18D20, 18D30
Theory and Applications of Categories, Vol. 28, 2013, No. 8, pp 213-240.