Completions in category theory: Difference between revisions
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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), |
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), |
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*'''free cocompletion''', '''free completion'''. These are obtained by freely adding colimits or limits. |
*'''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 [[presheaf (category theory)|presheaves]] on ''C''.<ref>Brian Day, Steve Lack, Limits of small functors, Journal of Pure and Applied Algebra, 210(3):651–683, 2007</ref><ref>https://ncatlab.org/nlab/show/free+cocompletion</ref> The free completion of ''C'' is the free cocompletion of the opposite of ''C''.<ref>https://ncatlab.org/nlab/show/free+completion</ref> |
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*'''Cauchy completion''' of a category ''C'' is roughly the closure of ''C'' in some ambient category so that all functors preserve limits.<ref>https://ncatlab.org/nlab/show/Cauchy+complete+category</ref> |
*'''Cauchy completion''' of a category ''C'' is roughly the closure of ''C'' in some ambient category so that all functors preserve limits.<ref>https://ncatlab.org/nlab/show/Cauchy+complete+category</ref> |
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*'''Isbell completion''' (also called reflexive completion), introduced by Isbell in 1960,<ref>J. R. Isbell. Adequate subcategories. Illinois Journal of Mathematics, 4:541–552, 1960.</ref> is in short the fixed-point category of the [[Isbell conjugacy]] adjunction.<ref>https://golem.ph.utexas.edu/category/2013/01/tight_spans_isbell_completions.html</ref><ref>{{harvnb|Avery|Leinster|2021}}</ref> It should not be confused with the [[Isbell envelop]], which was also introduced by Isbell. |
*'''Isbell completion''' (also called reflexive completion), introduced by Isbell in 1960,<ref>J. R. Isbell. Adequate subcategories. Illinois Journal of Mathematics, 4:541–552, 1960.</ref> is in short the fixed-point category of the [[Isbell conjugacy]] adjunction.<ref>https://golem.ph.utexas.edu/category/2013/01/tight_spans_isbell_completions.html</ref><ref>{{harvnb|Avery|Leinster|2021}}</ref> It should not be confused with the [[Isbell envelop]], which was also introduced by Isbell. |
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Revision as of 16:43, 16 July 2024
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.
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
- Avery, Tom; Leinster, Tom (2021), "Isbell conjugacy and the reflexive completion" (PDF), Theory and Applications of Categories, 36: 306–347, arXiv:2102.08290
Further reading
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