nLab free simplicial abelian group

Definition

The free simplicial abelian group functor

$$\mathbf{Z}[-]\colon sSet \to sAb$$

from SimplicialSets to SimplicialAbelianGroups is given by the functor

$$sSet = Fun(\Delta^{op}, Set) \to Fun(\Delta^{op}, Ab) = sAb,$$

where the middle functor applies the free abelian group functor

$$\mathbf{Z}[-]\colon Set \to Ab.$$

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.

Applications

Free simplicial abelian groups are the crucial ingredient of simplicial chains and simplicial cochains, and such also simplicial homology and simplicial cohomology?, in particular, singular homology and singular cohomology. See these articles for more information.

Related entries

• free abelian group

• simplicial chains

• simplicial cochains