Simplicial group explained

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 thatany simplicial abelian group

A is non-canonically homotopy equivalent to a product of Eilenberg–MacLane spaces, \prod _{i\geq} K(\pi _{iA,i).}

A commutative monoid in the category of simplicial abelian groups is a simplicial commutative ring.

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.

See also

• Simplicial commutative ring

References

• • Book: Goerss . P. G. . Jardine . J. F. . Simplicial Homotopy Theory . Birkhäuser . Basel, Boston, Berlin . Progress in Mathematics . 978-3-7643-6064-1 . 1999 . 174 .
• Charles Weibel, An introduction to homological algebra

External links

• • What is a simplicial commutative ring from the point of view of homotopy theory?