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 [math]\displaystyle{ A }[/math] is non-canonically homotopy equivalent to a product of Eilenberg–MacLane spaces, [math]\displaystyle{ \prod_{i\geq 0} K(\pi_iA,i). }[/math][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.
References
- Eckmann, Beno (1945), "Harmonische Funktionen und Randwertaufgaben in einem Komplex", Commentarii Mathematici Helvetici 17: 240–255, doi:10.1007/BF02566245
- Goerss, P. G.; Jardine, J. F. (1999). Simplicial Homotopy Theory. Progress in Mathematics. 174. Basel, Boston, Berlin: Birkhäuser. ISBN 978-3-7643-6064-1
- Charles Weibel, An introduction to homological algebra
External links
- simplicial group in nLab
- What is a simplicial commutative ring from the point of view of homotopy theory?
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