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| {{DISPLAYTITLE:''n''-skeleton}}
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| [[Image:Hypercubestar.svg|thumb|This [[hypercube graph]] is the {{nowrap|1-skeleton}} of the [[tesseract]].]]
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| :''This article is not about the [[topological skeleton]] concept of [[computer graphics]]''
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| In [[mathematics]], particularly in [[algebraic topology]], the {{nowrap|'''''n''-skeleton'''}} of a [[topological space]] ''X'' presented as a [[simplicial complex]] (resp. [[CW complex]]) refers to the [[subspace (topology)|subspace]] ''X''<sub>''n''</sub> that is the union of the simplices of ''X'' (resp. cells of ''X'') of dimensions {{nowrap|''m'' ≤ ''n''.}} In other words, given an inductive definition of a complex, the {{nowrap|''n''-skeleton}} is obtained by stopping at the {{nowrap|''n''-th step}}.
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| These subspaces increase with ''n''. The {{nowrap|0-skeleton}} is a [[discrete space]], and the {{nowrap|1-skeleton}} a [[topological graph]]. The skeletons of a space are used in [[obstruction theory]], to construct [[spectral sequence]]s by means of [[filtration]]s, and generally to make [[Mathematical induction|inductive argument]]s. They are particularly important when ''X'' has infinite dimension, in the sense that the ''X''<sub>''n''</sub> do not become constant as {{nowrap|''n'' → ∞.}}
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| == In geometry ==
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| In [[geometry]], a {{nowrap|''k''-skeleton}} of {{nowrap|''n''-[[polytope]]}} P (functionally represented as skel<sub>''k''</sub>(''P'')) consists of all {{nowrap|''i''-polytope}} elements of dimension up to ''k''.<ref>[[Peter McMullen]], Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. ISBN 0-521-81496-0 (Page 29)</ref>
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| For example:
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| : skel<sub>0</sub>(cube) = 8 vertices
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| : skel<sub>1</sub>(cube) = 8 vertices, 12 edges
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| : skel<sub>2</sub>(cube) = 8 vertices, 12 edges, 6 square faces
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| == For simplicial sets ==
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| The above definition of the skeleton of a simplicial complex is a particular case of the notion of skeleton of a [[simplicial set]]. Briefly speaking, a simplicial set <math>K_*</math> can be described by a collection of sets <math>K_i, \ i \geq 0</math>, together with face and degeneracy maps between them satisfying a number of equations. The idea of the ''n''-skeleton <math>sk_n(K_*)</math> is to first discard the sets <math>K_i</math> with <math>i > n</math> and then to complete the collection of the <math>K_i</math> with <math>i \leq n</math> to the "smallest possible" simplicial set so that the resulting simplicial set contains no non-degenerate simplices in degrees <math>i > n</math>.
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| More precisely, the restriction functor
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| :<math>i_*: \Delta^{op} Sets \rightarrow \Delta^{op}_{\leq n} Sets</math>
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| has a left adjoint, denoted <math>i^*</math>.<ref>{{Citation | last1=Goerss | first1=P. G. | last2=Jardine | first2=J. F. | title=Simplicial Homotopy Theory | publisher=Birkhäuser | location=Basel, Boston, Berlin | series=Progress in Mathematics | isbn=978-3-7643-6064-1 | year=1999 | volume=174}}, section IV.3.2</ref> (The notations <math>i^*, i_*</math> are comparable with the one of [[image functors for sheaves]].) The ''n''-skeleton of some simplicial set <math>K_*</math> is defined as
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| :<math>sk_n(K) := i^* i_* K.</math>
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| ===Coskeleton===
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| Moreover, <math>i_*</math> has a ''right'' adjoint <math>i^!</math>. The ''n''-coskeleton is defined as
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| :<math>cosk_n(K) := i^! i_* K.</math>
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| For example, the 0-skeleton of ''K'' is the constant simplicial set defined by <math>K_0</math>. The 0-coskeleton is given by the Cech [[nerve (category theory)|nerve]]
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| :<math>\dots \rightarrow K_0 \times K_0 \rightarrow K_0.</math>
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| (The boundary and degeneracy morphisms are given by various projections and diagonal embeddings, respectively.)
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| The above constructions work for more general categories (instead of sets) as well, provided that the category has [[fiber product]]s. The coskeleton is needed to define the concept of [[hypercovering]] in [[homotopical algebra]] and [[algebraic geometry]].<ref>{{Citation | last1=Artin | first1=Michael | author1-link=Michael Artin | last2=Mazur | first2=Barry | author2-link=Barry Mazur | title=Etale homotopy | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics, No. 100 | year=1969}}</ref>
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| == References ==
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| {{reflist}}
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| == External links ==
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| * {{MathWorld | urlname=Skeleton | title=Skeleton}}
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| [[Category:Algebraic topology]]
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| [[Category:General topology]]
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