nLab
Quillen adjunction between simplicial sets and connective dgc-algebras

Contents

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

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

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

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

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

Rational homotopy theory

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

(DiffGradedCommAlgebras k 0) proj op QuexpΩ PLdR SimplicialSets Qu \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 06:01:08. See the history of this page for a list of all contributions to it.