Completions in category theory
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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, 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
References
[edit]- ^ Brian Day, Steve Lack, Limits of small functors, Journal of Pure and Applied Algebra, 210(3):651–683, 2007
- ^ free cocompletion in nlab
- ^ free completion in nlab
- ^ Borceux & Dejean 1986
- ^ Cauchy complete category in nlab
- ^ Isbell 1960
- ^ Tight Spans, Isbell Completions and Semi-Tropical Modules, posted by Simon Willerton.
- ^ Avery & Leinster 2021
- ^ Karoubi envelope in nlab
- 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
- 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
- "Tight Spans, Isbell Completions and Semi-Tropical Modules", The n-Category Café
Further reading
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