Contents

model category

for ∞-groupoids

and

# Contents

## 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

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

## References

Last revised on September 25, 2020 at 13:11:34. See the history of this page for a list of all contributions to it.