N-skeleton

 ( Algebraic Topology ) 

In mathematics, particularly in algebraic topology, the of a topological space presented as a simplicial complex (resp. CW complex) refers to the subspace that is the union of the Simplex of (resp. cells of ) of dimensions In other words, given an inductive definition of a complex, the is obtained by stopping at the .

These subspaces increase with . The is a discrete space, and the a topological graph. The skeletons of a space are used in obstruction theory, to construct spectral sequences by means of filtrations, and generally to make inductive arguments. They are particularly important when has infinite dimension, in the sense that the do not become constant as

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In geometry In geometry, a of P (functionally represented as skel _k( P)) consists of all elements of dimension up to k.Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. (Page 29)

For example:

skel₀(cube) = 8 vertices

skel₁(cube) = 8 vertices, 12 edges

skel₂(cube) = 8 vertices, 12 edges, 6 square faces

The 1-skeleton is also known as the vertex-edge graph of the polytope.

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For simplicial sets 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 $K\_*$ can be described by a collection of sets $K\_i,\; \backslash \; i\; \backslash geq\; 0$, together with face and degeneracy maps between them satisfying a number of equations. The idea of the n-skeleton $sk\_n(K\_*)$ is to first discard the sets $K\_i$ with $i\; >\; n$ and then to complete the collection of the $K\_i$ with $i\; \backslash leq\; n$ to the "smallest possible" simplicial set so that the resulting simplicial set contains no non-degenerate simplices in degrees $i\; >\; n$.

More precisely, the restriction functor

$i\_*:\; \backslash Delta^\{op\}\; Sets\; \backslash rightarrow\; \backslash Delta^\{op\}\_\{\backslash leq\; n\}\; Sets$

has a left adjoint, denoted $i^*$., section IV.3.2 (The notations $i^*,\; i\_*$ are comparable with the one of image functors for sheaves.) The n-skeleton of some simplicial set $K\_*$ is defined as

$sk\_n(K)\; :=\; i^*\; i\_*\; K.$

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Coskeleton Moreover, $i\_*$ has a right adjoint $i^!$. The n-coskeleton is defined as

$cosk\_n(K)\; :=\; i^!\; i\_*\; K.$

For example, the 0-skeleton of K is the constant simplicial set defined by $K\_0$. The 0-coskeleton is given by the Cech nerve

$\backslash dots\; \backslash rightarrow\; K\_0\; \backslash times\; K\_0\; \backslash rightarrow\; K\_0.$

(The boundary and degeneracy morphisms are given by various projections and diagonal embeddings, respectively.)

The above constructions work for more general categories (instead of sets) as well, provided that the category has . The coskeleton is needed to define the concept of hypercovering in homotopical algebra and algebraic geometry.

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