Completions in category theory

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 :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. The free completion of C is the free cocompletion of the opposite of C. ind-completion. This is obtained by freely adding filtered colimits. Cauchy completion of a category C is roughly the closure of C in some ambient category so that all functors preserve limits. For example, if a metric space is viewed as an enriched category, then the Cauchy completion of it coincides with the usual completion of the space. Isbell completion, introduced by Isbell in 1960, is in short the fixed-point category of the Isbell conjugacy adjunction. 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. Exact completion