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: (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. 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.
• Isbell completion (also called reflexive 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 envelop, 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.

References

• Avery, Tom; Leinster, Tom (2021), "Isbell conjugacy and the reflexive completion" (PDF), Theory and Applications of Categories, 36: 306–347, arXiv:2102.08290
• Borceux, Francis; Dejean, Dominique (1986), "Cauchy completion in category theory", Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 (2): 133–146
• Isbell, J. R. (1960), "Adequate subcategories", Illinois Journal of Mathematics, 4 (4), doi:10.1215/ijm/1255456274

Further reading

• https://mathoverflow.net/questions/59291/completion-of-a-category