# Contents

## 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

There is a pair of adjoint functors (a free$\dashv$forgetful-adjunction)
$sAb \underoverset {\underset{ frgt }{\longrightarrow}} {\overset{\mathbb{Z}(-)}{\longleftarrow}} {\;\;\;\;\;\bot\;\;\;\;\;} sSet$