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

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 harvnb error: no target: CITEREFAveryLeinster2021 (help)
9. ↑ Karoubi envelope in nlab

## References

* Avery, Tom (2021), "Isbell conjugacy and the reflexive completion", Theory and Applications of Categories 36: 306–347, http://www.tac.mta.ca/tac/volumes/36/12/36-12.pdf
* Borceux, Francis; Dejean, Dominique (1986), "Cauchy completion in category theory", Cahiers de Topologie et Géométrie Différentielle Catégoriques 27 (2): 133–146, http://eudml.org/doc/91378
* 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, 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, https://ncatlab.org/nlab/show/free+completion
* free cocompletion, https://ncatlab.org/nlab/show/free+cocompletion
* Cauchy complete category, https://ncatlab.org/nlab/show/Cauchy+complete+category
* Karoubi envelope, https://ncatlab.org/nlab/show/Karoubi+envelope
* reflexive completion, https://ncatlab.org/nlab/show/reflexive+completion
* Willerton, Simon (2013), Tight Spans, Isbell Completions and Semi-Tropical Modules, https://golem.ph.utexas.edu/category/2013/01/tight_spans_isbell_completions.html

## 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)". https://www.youtube.com/watch?app=desktop&v=ycOb6_WVjOw.
* 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.

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