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- W2189349969 abstract "By a 'completion' on a 2-category K we mean here an idempotent pseu- domonad on K. We are particularly interested in pseudomonads that arise from KZ- doctrines. Motivated by a question of Lawvere, we compare the Cauchy completion (23), dened in the setting of V-Cat for V a symmetric monoidal closed category, with the Grothendieck completion (7), dened in the setting of S-Indexed Cat for S a topos. To this end we introduce a unied setting ('indexed enriched category theory') in which to formulate and study certain properties of KZ-doctrines. We nd 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 dierent ap- proach to answering the motivating question, that the Cauchy completion (dened using 'distributors' (2)) and the Grothendieck completion (dened using 'generalized functors' (21)) are actually equivalent constructions 1 ." @default.
- W2189349969 created "2016-06-24" @default.
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- W2189349969 date "2013-01-01" @default.
- W2189349969 modified "2023-09-26" @default.
- W2189349969 title "TIGHTLY BOUNDED COMPLETIONS" @default.
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