nLab
free simplicial abelian group

Contents

Contents

Definition

The free simplicial abelian group functor

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

from SimplicialSets to SimplicialAbelianGroups is given by the functor

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

where the middle functor applies the free abelian group functor

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

Properties

Proposition

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

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

between SimplicialAbelianGroups and SimplicialSets, where

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.

Last revised on July 12, 2021 at 13:35:57. See the history of this page for a list of all contributions to it.