In category theory, a branch of mathematics, there are several ways (completions) to enlarge a given category in a way somehow analogous to a completion in topology. These are (ignoring the set-theoretic matters for simplicity):
free cocompletion, free completion. These are obtained by freely adding colimits or limits. Explicitly, the free cocompletion of a category C is the Yoneda embedding of C into the category of presheaves on C.[1][2] The free completion of C is the free cocompletion of the opposite of C.[3]
Cauchy completion of a category C is roughly the closure of C in some ambient category so that all functors preserve limits.[4][5] For example, if a metric space is viewed as an enriched category (see generalized metric space), then the Cauchy completion of it coincides with the usual completion of the space.
Isbell completion (also called reflexive completion), introduced by Isbell in 1960,[6] is in short the fixed-point category of the Isbell conjugacy adjunction.[7][8] It should not be confused with the Isbell envelope, which was also introduced by Isbell.
Karoubi envelope or idempotent completion of a category C is (roughly) the universal enlargement of C so that every idempotent is a split idempotent.[9]
Borceux, Francis; Dejean, Dominique (1986), "Cauchy completion in category theory", Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 (2): 133–146
Carboni, A.; Vitale, E.M. (1998), "Regular and exact completions", Journal of Pure and Applied Algebra, 125 (1–3): 79–116, doi:10.1016/S0022-4049(96)00115-6
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