category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, a branch of mathematics, there are several ways (completions) to enlarge a given category in a way somehow analogous to a completion in

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

. 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 In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewi ...

of ''C'' into the category of

presheaves In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...

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 colimit In category theory, filtered categories generalize the notion of directed set understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered category). There is a dual notion of cofiltered c ...

s. *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 (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, is in short the fixed-point category of the

Isbell conjugacy Isbell conjugacy (a.k.a. Isbell duality or Isbell adjunction) (named after John R. Isbell) is a fundamental construction of enriched category theory formally introduced by William Lawvere in 1986. That is a duality between covariant and contravaria ...

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 In category theory, a branch of mathematics, the exact completion constructs a Barr-exact category from any finitely complete category. It is used to form the effective topos and other realizability toposes. Construction Let ''C'' be a category w ...

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