# nLab Quillen adjunction between simplicial sets and connective dgc-algebras

Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of $(\infty,1)$-categories

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

for stable $(\infty,1)$-categories

for $(\infty,1)$-operads

for $(n,r)$-categories

for $(\infty,1)$-sheaves / $\infty$-stacks

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 23, 2021 at 10:01:08. See the history of this page for a list of all contributions to it.