Editing Simplicial set (section)

==Simplicial objects==

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

:''X'' : Δ → ''C''

or equivalently a covariant functor
:''X'': Δ<sup>op</sup> → ''C,''

where Δ still denotes the [[simplex category]] and <sup>op</sup> the [[opposite category]]. When ''C'' is the [[category of sets]], we are just talking about the simplicial sets that were defined above.  Letting ''C'' be the [[category of groups]] or [[category of abelian groups]], we obtain the categories '''sGrp''' of simplicial [[group (mathematics)|group]]s and '''sAb''' of simplicial [[abelian group]]s, respectively.

[[Simplicial group]]s 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 complex]]es and is given by functors

:''N:'' '''sAb''' → Ch<sub>+</sub>

and

: &Gamma;: Ch<sub>+</sub> → &nbsp;'''sAb'''.

See also: [[simplicial diagram]].