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. 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 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

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
• 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
• Day, Brian J.; Lack, Stephen (2007), "Limits of small functors", Journal of Pure and Applied Algebra, 210 (3): 651–663, arXiv:math/0610439, doi:10.1016/j.jpaa.2006.10.019
• Isbell, J. R. (1960), "Adequate subcategories", Illinois Journal of Mathematics, 4 (4), doi:10.1215/ijm/1255456274
• "free completion", ncatlab.org
• "free cocompletion", ncatlab.org
• "Cauchy complete category", ncatlab.org
• "Karoubi envelope", ncatlab.org
• "reflexive completion", ncatlab.org
• Willerton, Simon (2013), "Tight Spans, Isbell Completions and Semi-Tropical Modules", The n-Category Café, arXiv:1302.4370

Further reading

• https://mathoverflow.net/questions/59291/completion-of-a-category
• Leinster, Tom (28 February 2022). "Leinster - The functoriality of the reflexive completion (Category Theory 20→21)". YouTube.
• Carboni, A. (15 September 1995). "Some free constructions in realizability and proof theory". Journal of Pure and Applied Algebra. 103 (2): 117–148. doi:10.1016/0022-4049(94)00103-p.