Simplicial group

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, more precisely, in the theory of <a href="/facts/Simplicial_set/6DaqXMLT">simplicial sets</a>, a simplicial group is a <a href="/facts/Simplicial_object/6DaqXMLT">simplicial object</a> in the <a href="/facts/Category_of_groups/sxSd9Kbc">category of groups</a>. Similarly, a simplicial abelian group is a simplicial object in the <a href="/facts/Category_of_abelian_groups/jF3sXRFn">category of abelian groups</a>. A simplicial group is a <a href="/facts/Kan_complex/TWxx0fWK">Kan complex</a> (in particular, its <a href="/facts/Homotopy_group/xjogL95Y">homotopy groups</a> make sense). The <a href="/facts/Dold%E2%80%93Kan_correspondence/7BSSupVt">Dold–Kan correspondence</a> says that a simplicial abelian group may be identified with a <a href="/facts/Chain_complex/YAlchtCT">chain complex</a>. In fact it can be shown that
any simplicial abelian group 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is non-canonically <a href="/facts/Homotopy_equivalent/1nWrPxdj">homotopy equivalent</a> to a product of <a href="/facts/Eilenberg%E2%80%93MacLane_space/mrux6oMJ">Eilenberg–MacLane spaces</a>, 
 
 
 
 
 ∏
 
 i
 ≥
 0
 
 
 K
 (
 
 π
 
 i
 
 
 A
 ,
 i
 )
 .
 
 
 {\displaystyle \prod _{i\geq 0}K(\pi _{i}A,i).}

A <a href="/facts/Commutative_monoid/6AHve7p8">commutative monoid</a> in the <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> of simplicial abelian groups is a <a href="/facts/Simplicial_commutative_ring/6LktuDH8">simplicial commutative ring</a>.
Eckmann (1945) discusses a simplicial analogue of the fact that a <a href="/facts/Cohomology_class/J7FJzTNh">cohomology class</a> on a <a href="/facts/K%C3%A4hler_manifold/JFPM6Vxg">Kähler manifold</a> has a unique <a href="/facts/Harmonic_function/H6pg4dMT">harmonic representative</a> and deduces <a href="/facts/Kirchhoff%27s_circuit_laws/fGAAFXB3">Kirchhoff's circuit laws</a> from these observations.

Simplicial group open-in-new

Simplicial group