In mathematics, more precisely, in the theory of simplicial sets, a simplicial group is a simplicial object in the category of groups. Similarly, a simplicial abelian group is a simplicial object in the category of abelian groups. A simplicial group is a Kan complex (in particular, its homotopy groups make sense). The Dold–Kan correspondence says that a simplicial abelian group may be identified with a chain complex. In fact it can be shown that
any simplicial abelian group
A
{\displaystyle A}
is non-canonically homotopy equivalent to a product of Eilenberg–MacLane spaces,
∏
i
≥
0
K
(
π
i
A
,
i
)
.
{\displaystyle \prod _{i\geq 0}K(\pi _{i}A,i).}
A commutative monoid in the category of simplicial abelian groups is a simplicial commutative ring.
Eckmann (1945) discusses a simplicial analogue of the fact that a cohomology class on a Kähler manifold has a unique harmonic representative and deduces Kirchhoff's circuit laws from these observations.
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