Skeleton (category theory) 1812151 218916519 2008-06-12T19:19:37Z Apokrif 173030 {{otheruses|Skeleton (disambiguation)}} {{otheruses|Skeleton (disambiguation)}} 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. == Definition == 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: *Every object of ''D'' is an object of ''C''. *(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. :<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''. *The composition law in ''D'' is the restriction of the composition law in ''C'' to the morphisms in ''D''. *(Isomorphic-dense) Every ''C''-object is isomorphic to some ''D''-object. *No two distinct ''D''-objects are isomorphic. == Existence and uniqueness == 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]]. 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. == Examples == *The category [[category of sets|'''Set''']] of all [[set]]s has the subcategory of all [[cardinal number]]s as a skeleton. *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 category of all [[well-order|well-ordered sets]] has the subcategory of all [[ordinal numbers]] as a skeleton. ==References== * Adámek, Jiří, Herrlich, Horst, & Strecker, George E. (1990). [http://www.math.uni-bremen.de/~dmb/acc.pdf ''Abstract and Concrete Categories'']. Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition) [[Category:Category theory]] [[de:Skelett (Kategorientheorie)]]