Completions in category theory
Appearance
In category theory, a branch of mathematics, there are several ways (completions) to enlarge a given category in a way similar to complete a topological space like a metric space. 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]
- Cauchy completion of a category C is roughly the closure of C in some ambient category so that all functors preserve limits.[4]
- Isbell completion (also called reflexive completion), introduced by Isbell in 1960,[5] is in short the fixed-point category of the Isbell conjugacy adjunction.[6][7] 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.[8]
References
- ^ Brian Day, Steve Lack, Limits of small functors, Journal of Pure and Applied Algebra, 210(3):651–683, 2007
- ^ https://ncatlab.org/nlab/show/free+cocompletion
- ^ https://ncatlab.org/nlab/show/free+completion
- ^ https://ncatlab.org/nlab/show/Cauchy+complete+category
- ^ J. R. Isbell. Adequate subcategories. Illinois Journal of Mathematics, 4:541–552, 1960.
- ^ https://golem.ph.utexas.edu/category/2013/01/tight_spans_isbell_completions.html
- ^ Avery & Leinster 2021
- ^ https://ncatlab.org/nlab/show/Karoubi+envelope
- Avery, Tom; Leinster, Tom (2021), "Isbell conjugacy and the reflexive completion" (PDF), Theory and Applications of Categories, 36: 306–347, arXiv:2102.08290
- Francis Borceux and D. Dejean, Cauchy completion in category theory Cahiers Topologie Géom. Différentielle Catégoriques, 27:133–146, (1986)