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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, $\prod _{i\geq 0}K(\pi _{i}A,i).$ ^[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

^ Paul Goerss and Rick Jardine (1999, Ch 3. Proposition 2.20)

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

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[1] Paul Goerss and Rick Jardine (1999, Ch 3. Proposition 2.20)

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-In mathematics, more precisely, in the theory of [[simplicial set]]s, 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.<!-- need more precision --> In fact it can be shown that
+In mathematics, more precisely, in the theory of [[simplicial set]]s, 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]].<!-- need more precision --> In fact it can be shown that
-any simplicial abelian group <math>A</math> is non-canonically homotopy equivalent to a product of [[Eilenberg–MacLane space]]s, <math>\prod_{i\geq 0} K(\pi_iA,i).</math><ref>{{harvs|txt|last1=Goerss|first1=Paul|last2=Jardine|first2=Rick|authorlink2=Rick Jardine|year=1999|loc=Ch 3. Proposition 2.20}}</ref>
+any simplicial abelian group <math>A</math> is non-canonically [[homotopy equivalent]] to a product of [[Eilenberg–MacLane space]]s, <math>\prod_{i\geq 0} K(\pi_iA,i).</math><ref>{{harvs|txt|last1=Goerss|first1=Paul|last2=Jardine|first2=Rick|authorlink2=Rick Jardine|year=1999|loc=Ch 3. Proposition 2.20}}</ref>
-A commutative monoid in the category of simplicial abelian groups is a [[simplicial commutative ring]].
+A [[commutative monoid]] in the category of simplicial abelian groups is a [[simplicial commutative ring]].
-{{harvtxt|Eckmann|1945}} discusses a simplicial analogue of the fact that a cohomology class on a [[Kähler manifold]] has a unique [[harmonic function|harmonic representative]] and deduces [[Kirchhoff's circuit laws]] from these observations.
+{{harvtxt|Eckmann|1945}} discusses a simplicial analogue of the fact that a [[cohomology class]] on a [[Kähler manifold]] has a unique [[harmonic function|harmonic representative]] and deduces [[Kirchhoff's circuit laws]] from these observations.
 == References ==
 {{reflist}}
-* {{Citation|last=Eckmann|first=Beno|title=Harmonische Funktionen und Randwertaufgaben in einem Komplex|journal=Commentarii Mathematici Helvetici|volume=17|year=1945|pages=240&ndash;255|mr=0013318|doi=10.1007/BF02566245}}
+* {{Citation|last=Eckmann|first=Beno|authorlink = Beno Eckmann|title=Harmonische Funktionen und Randwertaufgaben in einem Komplex|journal=[[Commentarii Mathematici Helvetici]]|volume=17|year=1945|pages=240&ndash;255|mr=0013318|doi=10.1007/BF02566245}}
 * {{Cite book | last1=Goerss | first1=P. G. | last2=Jardine | first2=J. F. | title=Simplicial Homotopy Theory | publisher=Birkhäuser | location=Basel, Boston, Berlin | series=Progress in Mathematics | isbn=978-3-7643-6064-1 | year=1999 | volume=174 | postscript=<!--None-->}}
 * [[Charles Weibel]], ''An introduction to homological algebra''