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, $\prod_{i\geq 0} K(\pi_iA,i).$ {{harvs|txt|last1=Goerss|first1=Paul|last2=Jardine|first2=Rick|authorlink2=Rick Jardine|year=1999|loc=Ch 3. Proposition 2.20}}

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 representative and deduces Kirchhoff's circuit laws from these observations.

References

{{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=240–255|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 }}
Charles Weibel, An introduction to homological algebra

Simplicial group

See also

References

Further reading