N-skeleton


In mathematics, particularly in algebraic topology, the of a topological space X presented as a simplicial complex refers to the subspace Xn that is the union of the simplices of X of dimensions In other words, given an inductive definition of a complex, the is obtained by stopping at the.
These subspaces increase with n. 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 X has infinite dimension, in the sense that the Xn do not become constant as

In geometry

In geometry, a of P consists of all elements of dimension up to k.
For example:

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 can be described by a collection of sets, together with face and degeneracy maps between them satisfying a number of equations. The idea of the n-skeleton is to first discard the sets with and then to complete the collection of the with to the "smallest possible" simplicial set so that the resulting simplicial set contains no non-degenerate simplices in degrees.
More precisely, the restriction functor
has a left adjoint, denoted. The n-skeleton of some simplicial set is defined as

Coskeleton

Moreover, has a right adjoint. The n-coskeleton is defined as
For example, the 0-skeleton of K is the constant simplicial set defined by. The 0-coskeleton is given by the Cech nerve
The above constructions work for more general categories 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.