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 simplices 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 argumentss. They are particularly important when has infinite dimension, in the sense that the do not become constant as

In geometry

In geometry, a of P (functionally represented as skelk(P)) consists of all elements of dimension up to k.

For example:

1. skel0(cube) = 8 vertices

2. skel1(cube) = 8 vertices, 12 edges

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

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

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, together with face and degeneracy maps between them satisfying a number of equations. The idea of the n.

More precisely, the restriction functor

. are comparable with the one of image functors for sheaves.) The n is defined as

Coskeleton

has a right. The n-coskeleton is defined as

For example, the 0-skeleton of K. The 0-coskeleton is given by the Čech nerve

(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 fiber products. The coskeleton is needed to define the concept of hypercovering in homotopical algebra and algebraic geometry.

References

External links