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| {{other uses|Skeleton (disambiguation)}}
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| In [[mathematics]], a '''skeleton''' of a [[category (category theory)|category]] is a [[subcategory]] which, roughly speaking, does not contain any extraneous [[isomorphism]]s. In a certain sense, the skeleton of a category is the "smallest" [[equivalence of categories|equivalent]] category which captures all "categorical properties". In fact, two categories are [[equivalence of categories|equivalent]] [[iff|if and only if]] they have [[isomorphism of categories|isomorphic]] skeletons. A category is called '''skeletal''' if isomorphic objects are necessarily identical.
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| == Definition ==
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| A skeleton of a category ''C'' is a [[subcategory|full]], isomorphism-dense [[subcategory]] ''D'' in which no two distinct objects are isomorphic. In detail, a skeleton of ''C'' is a category ''D'' such that:
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| *Every object of ''D'' is an object of ''C''.
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| *(Fullness) For every pair of objects ''d''<sub>1</sub> and ''d''<sub>2</sub> of ''D'', the [[morphism]]s in ''D'' are precisely the morphisms in ''C'', i.e.
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| :<math>hom_D(d_1, d_2) = hom_C(d_1, d_2)</math> | |
| *For every object ''d'' of ''D'', the ''D''-identity on ''d'' is the ''C''-identity on ''d''.
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| *The composition law in ''D'' is the restriction of the composition law in ''C'' to the morphisms in ''D''.
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| *(Isomorphism-dense) Every ''C''-object is isomorphic to some ''D''-object.
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| *No two distinct ''D''-objects are isomorphic.
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| == Existence and uniqueness ==
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| It is a basic fact that every small category has a skeleton; more generally, every [[accessible category]] has a skeleton. (This is equivalent to the [[axiom of choice]].) Also, although a category may have many distinct skeletons, any two skeletons are [[isomorphism of categories|isomorphic as categories]], so [[up to]] isomorphism of categories, the skeleton of a category is [[unique]].
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| The importance of skeletons comes from the fact that they are (up to isomorphism of categories), canonical representatives of the equivalence classes of categories under the [[equivalence relation]] of [[equivalence of categories]]. This follows from the fact that any skeleton of a category ''C'' is [[equivalence of categories|equivalent]] to ''C'', and that two categories are equivalent if and only if they have isomorphic skeletons.
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| == Examples ==
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| *The category '''[[category of sets|Set]]''' of all [[Set (mathematics)|sets]] has the subcategory of all [[cardinal number]]s as a skeleton.
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| *The category '''[[category of vector spaces|K-Vect]]''' of all [[vector space]]s over a fixed [[field (mathematics)|field]] <math>K</math> has the subcategory consisting of all powers <math>K^n</math>, where ''n'' is any cardinal number, as a skeleton; the maps <math>K^m \to K^n</math> are exactly the ''n''×''m'' [[Matrix (mathematics)|matrices]] with entries in ''K''.
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| *'''[[FinSet]]''', the category of all [[finite set]]s has '''[[FinOrd]]''', the category of all finite [[ordinal numbers]], as a skeleton.
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| *The category of all [[well-order|well-ordered sets]] has the subcategory of all [[ordinal numbers]] as a skeleton.
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| *A [[preorder]], i.e. a small category such that for every pair of objects <math> A,B </math>, the set <math> Hom(A,B)</math> either has one element or is empty, has a [[partially ordered set]] as a skeleton.
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| ==References==
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| * Adámek, Jiří, Herrlich, Horst, & Strecker, George E. (1990). [http://katmat.math.uni-bremen.de/acc/acc.pdf ''Abstract and Concrete Categories'']. Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition)
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| * Robert Goldblatt (1984). ''Topoi, the Categorial Analysis of Logic'' (Studies in logic and the foundations of mathematics, 98). North-Holland. Reprinted 2006 by Dover Publications.
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| [[Category:Category theory]]
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