nLab
Quillen adjunction between simplicial sets and connective dgc-algebras

Statement

Proposition

(Quillen adjunction between simplicial sets and connective dgc-algebras)

The PL de Rham complex-construction is the left adjoint in a Quillen adjunction between

• the opposite of the projective model structure on connective dgc-algebras

• the classical model structure on simplicial sets

$$\big( DiffGradedCommAlgebras^{\geq 0}_{k} \big)^{op}_{proj} \underoverset { \underset {\;\;\; exp \;\;\;} {\longrightarrow} } { \overset {\;\;\;\Omega^\bullet_{PLdR}\;\;\;} {\longleftarrow} } {\bot_{\mathrlap{Qu}}} SimplicialSets_{Qu}$$

Proof

That the PL de Rham complex functor preserves cofibrations, hence sends injections of simplicial sets to surjections of dgc-algebras, is immediate from its construction.

That its right adjoint preserves fibrations, hence sends cofibrations of dgc-algebras to Kan fibrations, is the statement of Bousfield-Gugenheim 76, Lemma 8.2.

Related statements

• fundamental theorem of dg-algebraic rational homotopy theory

• Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras

References

• Aldridge Bousfield, Victor Gugenheim, On PL deRham theory and rational homotopy type, Memoirs of the AMS, vol. 179 (1976) (ams:memo-8-179)