Simplicial group

It has been suggested that Simplicial commutative ring be merged into this article. (Discuss) Proposed since July 2024.

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 ${\displaystyle A}$ is non-canonically homotopy equivalent to a product of Eilenberg–MacLane spaces, ${\displaystyle \prod _{i\geq 0}K(\pi _{i}A,i).}$^[1]

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.