nLab
fundamental theorem of dg-algebraic rational homotopy theory

Contents

Contents

Idea

The fundamental theorem of rational homotopy theory modeled by dgc-algebras.

Preliminaries

Definition

(nilpotent and finite rational homotopy types)

Write

(1)Ho(SimplicialSets Qu) 1,nil fin AAAHo(SimplicialSets Qu) Ho \big( SimplicialSets_{Qu} \big)^{fin_{\mathbb{Q}}}_{\geq 1, nil} \overset{ \phantom{AAA} }{\hookrightarrow} Ho \big( SimplicialSets_{Qu} \big)

for the full subcategory of the classical homotopy category (homotopy category of the classical model structure on simplicial sets) on those homotopy types XX which are

  • connected: π 0(X)=*\pi_0(X) = \ast

  • nilpotent: π 1(X)\pi_1(X) is a nilpotent group

  • rational finite type: dim (H n(X;,))<dim_{\mathbb{Q}}\big( H^n(X;,\mathbb{Q}) \big) \lt \infty for all nn \in \mathbb{N}.

and

(2)Ho(SimplicialSets Qu) 1,nil ,fin AAAHo(SimplicialSets Qu) Ho \big( SimplicialSets_{Qu} \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil} \overset{ \phantom{AAA} }{\hookrightarrow} Ho \big( SimplicialSets_{Qu} \big)

for the further full subcategory on those homotopy types that are already rational.

Similarly, write

(3)Ho(DiffGradedCommAlgebras 0) fin 1AAAHo(DiffGradedCommAlgebras 0) Ho \big( DiffGradedCommAlgebras^{\geq 0}_{\mathbb{Q}} \big)_{fin}^{\geq 1} \overset{ \phantom{AAA} }{\hookrightarrow} Ho \big( DiffGradedCommAlgebras^{\geq 0}_{\mathbb{Q}} \big)

for the full subcategory of the homotopy category of the projective model structure on connective dgc-algebras on those dgc-algebras AA which are

  • connected: H 0(A)H^0(A) \simeq \mathbb{Q}

  • finite type: dim (H n(A))<dim_{\mathbb{Q}}\big( H^n(A) \big) \lt \infty for all nn \in \mathbb{N}.

(Bousfield-Gugenheim 76, 9.2)

Statement

Proposition

(fundamental theorem of dg-algebraic rational homotopy theory)

The derived adjunction

Ho((DiffGradedCommAlgebras k 0) proj op)exp𝕃Ω PLdR Ho(SimplicialSets Qu) Ho \left( \big( DiffGradedCommAlgebras^{\geq 0}_{k} \big)^{op}_{proj} \right) \underoverset { \underset {\;\;\; \mathbb{R} exp \;\;\;} {\longrightarrow} } { \overset {\;\;\; \mathbb{L} \Omega^\bullet_{PLdR}\;\;\;} {\longleftarrow} } {\bot} Ho \big( SimplicialSets_{Qu} \big)

of the Quillen adjunction between simplicial sets and connective dgc-algebras (whose left adjoint is the PL de Rham complex-functor) has the following properties:

  • on connected, nilpotent rationally finite homotopy types XX (1) the derived adjunction unit is rationalization

    Ho(SimplicialSets Qu) 1,nil fin Ho(SimplicialSets Qu) 1,nil ,fin X expΩ PLdR (X) \array{ Ho \big( SimplicialSets_{Qu} \big)^{fin_{\mathbb{Q}}}_{\geq 1, nil} & \overset{ }{\longrightarrow} & Ho \big( SimplicialSets_{Qu} \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil} \\ X &\mapsto& \mathbb{R}\exp \circ \Omega^\bullet_{PLdR}(X) }
    Xη X derrationalizationexpΩ PLdR (X) X \underoverset {\eta_X^{der}} {rationalization} {\longrightarrow} \mathbb{R}\exp \circ \Omega^\bullet_{PLdR}(X)
  • on the full subcategories of nilpotent and finite rational homotopy types from Def. it restricts to an equivalence of categories:

    Ho((DiffGradedCommAlgebras k 0) proj op) fin 1exp𝕃Ω PLdR Ho(SimplicialSets Qu) 1,nil ,fin Ho \left( \big( DiffGradedCommAlgebras^{\geq 0}_{k} \big)^{op}_{proj} \right)^{\geq 1}_{fin} \underoverset { \underset {\;\;\; \mathbb{R} exp \;\;\;} {\longrightarrow} } { \overset {\;\;\; \mathbb{L} \Omega^\bullet_{PLdR}\;\;\;} {\longleftarrow} } {\simeq} Ho \big( SimplicialSets_{Qu} \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil}

(Bousfield-Gugenheim 76, Theorems 9.4 & 11.2)

References

Last revised on December 22, 2020 at 22:52:44. See the history of this page for a list of all contributions to it.