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.{{harvnb|Day|Lack|2007}}{{harvnb|free cocompletion in nlab}} The free completion of C is the free cocompletion of the opposite of C.{{harvnb|free completion in nlab}}
- 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.{{harvnb|Borceux|Dejean|1986}}{{harvnb|Cauchy complete category in nlab}} 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,{{harvnb|Isbell|1960}} is in short the fixed-point category of the Isbell conjugacy adjunction.{{harvnb|Tight Spans, Isbell Completions and Semi-Tropical Modules, posted by Simon Willerton.}}{{harvnb|Avery|Leinster|2021}} 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.{{harvnb|Karoubi envelope in nlab}}
- Exact completion
Notes
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References
- {{citation
|url=http://www.tac.mta.ca/tac/volumes/36/12/36-12.pdf
|arxiv=2102.08290
|last1=Avery |first1=Tom
|last2=Leinster |first2=Tom
|title=Isbell conjugacy and the reflexive completion
|journal=Theory and Applications of Categories
|date=2021
|volume=36
|pages=306–347}}
- {{citation |url=http://eudml.org/doc/91378 |title=Cauchy completion in category theory |journal=Cahiers de Topologie et Géométrie Différentielle Catégoriques |date=1986 |volume=27 |issue=2 |pages=133–146 |last1=Borceux |first1=Francis |last2=Dejean |first2=Dominique }}
- {{citation|doi=10.1016/S0022-4049(96)00115-6 |title=Regular and exact completions |date=1998 |last1=Carboni |first1=A. |last2=Vitale |first2=E.M. |journal=Journal of Pure and Applied Algebra |volume=125 |issue=1–3 |pages=79–116 }}
- {{citation |doi=10.1016/j.jpaa.2006.10.019 |title=Limits of small functors |date=2007 |last1=Day |first1=Brian J. |last2=Lack |first2=Stephen |journal=Journal of Pure and Applied Algebra |volume=210 |issue=3 |pages=651–663 |arxiv=math/0610439 }}
- {{citation |doi=10.1215/ijm/1255456274 |title=Adequate subcategories |date=1960 |last1=Isbell |first1=J. R. |journal=Illinois Journal of Mathematics |volume=4 |issue=4 }}
- {{citation |title=free completion |url=https://ncatlab.org/nlab/show/free+completion |website=ncatlab.org|ref={{harvid|free completion in nlab}}}}
- {{citation |title=free cocompletion |url=https://ncatlab.org/nlab/show/free+cocompletion |website=ncatlab.org|ref={{harvid|free cocompletion in nlab}}}}
- {{citation |title=Cauchy complete category |url=https://ncatlab.org/nlab/show/Cauchy+complete+category |website=ncatlab.org|ref={{harvid|Cauchy complete category in nlab}}}}
- {{citation |title=Karoubi envelope |url=https://ncatlab.org/nlab/show/Karoubi+envelope |website=ncatlab.org|ref={{harvid|Karoubi envelope in nlab}}}}
- {{citation |title=reflexive completion |url=https://ncatlab.org/nlab/show/reflexive+completion |website=ncatlab.org|ref={{harvid|reflexive completion in nlab}}}}
- {{citation |title=Tight Spans, Isbell Completions and Semi-Tropical Modules |url=https://golem.ph.utexas.edu/category/2013/01/tight_spans_isbell_completions.html |website=The n-Category Café|ref={{harvid|Tight Spans, Isbell Completions and Semi-Tropical Modules, posted by Simon Willerton.}} |last1=Willerton |first1=Simon |date=2013 |arxiv=1302.4370 }}
External links
- https://mathoverflow.net/questions/59291/completion-of-a-category
- {{cite web |last1=Leinster |first1=Tom |title=Leinster - The functoriality of the reflexive completion (Category Theory 20→21) |website=YouTube |url=https://www.youtube.com/watch?app=desktop&v=ycOb6_WVjOw |date=28 February 2022}}