N-skeleton

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 ${\displaystyle K_{\ast }}$ can be described by a collection of sets ${\displaystyle K_i,i\ge 0}$, together with face and degeneracy maps between them satisfying a number of equations. The idea of the n-skeleton ${\displaystyle s{k}_n(K_{\ast })}$ is to first discard the sets ${\displaystyle K_i}$ with ${\displaystyle i>n}$ and then to complete the collection of the ${\displaystyle K_i}$ with ${\displaystyle i\le n}$ to the "smallest possible" simplicial set so that the resulting simplicial set contains no non-degenerate simplices in degrees ${\displaystyle i>n}$.

More precisely, the restriction functor

${\displaystyle i_{\ast }:{\mathrm{\Delta }}^{op}Sets\to {\mathrm{\Delta }}_{\le n}^{op}Sets}$

has a left adjoint, denoted ${\displaystyle i^{\ast }}$. (The notations ${\displaystyle i^{\ast },i_{\ast }}$ are comparable with the one of image functors for sheaves.) The n-skeleton of some simplicial set ${\displaystyle K_{\ast }}$ is defined as

${\displaystyle s{k}_n(K):=i^{\ast }{i}_{\ast }K.}$

Coskeleton

Moreover, ${\displaystyle i_{\ast }}$ has a right adjoint ${\displaystyle i^{!}}$. The n-coskeleton is defined as

${\displaystyle cos{k}_n(K):=i^{!}{i}_{\ast }K.}$

For example, the 0-skeleton of K is the constant simplicial set defined by ${\displaystyle K_0}$. The 0-coskeleton is given by the Cech nerve

${\displaystyle \cdots \to K_0\times K_0\to K_0.}$

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