Simplicial Set - Simplicial Objects

Simplicial Objects

A simplicial object X in a category C is a contravariant functor

X: Δ → C

or equivalently a covariant functor

X: Δop → C

When C is the category of sets, we are just talking about simplicial sets. Letting C be the category of groups or category of abelian groups, we obtain the categories sGrp of simplicial groups and sAb of simplicial abelian groups, respectively.

Simplicial groups and simplicial abelian groups also carry closed model structures induced by that of the underlying simplicial sets.

The homotopy groups of simplicial abelian groups can be computed by making use of the Dold-Kan correspondence which yields an equivalence of categories between simplicial abelian groups and bounded chain complexes and is given by functors

N: sAb → Ch₊

and

Γ: Ch₊ → sAb.