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.[1][2] The free completion of C is the free cocompletion of the opposite of C.[3]
* 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.[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]
* Exact completion

## Notes

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1. ^ Day & Lack 2007
2. ^ free cocompletion in nlab
3. ^ free completion in nlab
4. ^ Borceux & Dejean 1986
5. ^ Cauchy complete category in nlab
6. ^ Isbell 1960
7. ^ Tight Spans, Isbell Completions and Semi-Tropical Modules, posted by Simon Willerton.
8. ^ Avery & Leinster 2021
9. ^ Karoubi envelope in nlab

## References

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* 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)00115-61–3&rft.pages=79-116&rft.date=1998&rft_id=info:doi/10.1016/S0022-4049(96)00115-6&rft.aulast=Carboni&rft.aufirst=A.&rft.au=Vitale,+E.M.&rfr_id=info:sid/en.wikipedia.org:Completions+in+category+theory" class="Z3988">
* 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

## External links

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* 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.