nLab simplicial abelian group

Definition

A simplicial abelian group is

• an abelian group object internal to SimplicialSets;

• a simplicial object internal to AbelianGroups.

Properties

Basic properties

Proposition

(free simplicial abelian group-adjunction)
There is a pair of adjoint functors (a free$\dashv$forgetful-adjunction)

$$sAb \underoverset {\underset{ frgt }{\longrightarrow}} {\overset{\mathbb{Z}(-)}{\longleftarrow}} {\;\;\;\;\;\bot\;\;\;\;\;} sSet$$

between SimplicialAbelianGroups and SimplicialSets, where

• the left adjoint is the free simplicial abelian group-functor;

• the right adjoint is the functor that assigns underlying simplicial sets.

This is a Quillen adjunction with respect to the classical model structure on simplicial sets and the projective model structure on simplicial abelian groups.

More properties

• Dold-Kan correspondence

• Eilenberg-Zilber theorem

• model structure on simplicial abelian groups

• cosimplicial abelian group

Related concepts

• Moore complex

• Eilenberg-Zilber map

• simplicial group, simplicial topological group

• free simplicial abelian group