Simplicial group

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

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.

See also

• Simplicial commutative ring

References

• Eckmann, Beno (1945), "Harmonische Funktionen und Randwertaufgaben in einem Komplex", Commentarii Mathematici Helvetici, 17: 240–255, doi:10.1007/BF02566245, MR 0013318
• Goerss, P. G.; Jardine, J. F. (1999). Simplicial Homotopy Theory. Progress in Mathematics. Vol. 174. Basel, Boston, Berlin: Birkhäuser. ISBN 978-3-7643-6064-1.
• Charles Weibel, An introduction to homological algebra

Further reading

• simplicial group at the nLab
• What is a simplicial commutative ring from the point of view of homotopy theory?
• W. G. Dwyer and D. M. Kan. “Homotopy theory and simplicial groupoids”. In: Nederl. Akad. Wetensch. Indag. Math. 46.4 (1984), pp. 379–385.
• http://pantodon.jp/index.rb?body=simplicial_group in Japanese