rationalization in nLab

Context

Rational homotopy theory

• Idea
• Definition
• Rationalization of a homotopy type
• Rationalization of simply-connected spaces
• π 1\pi_1-Rationalization of connected spaces
• Rationalization as a localization of TopTop/∞Grpd\infty Grpd
• In homotopy type theory
• Properties
• Rationalization via PL de Rham theory
• Preservation of homotopy pullbacks
• Preservation of homotopy fibers
• Rationalization of spectra
• Related concepts
• References

Definition

(π 1\pi_1-rationalization)
A map η X ℚ:X⟶X ℚ\eta^{\mathbb{Q}}_X \;\colon\; X \longrightarrow X_{\mathbb{Q}} of connected homotopy types is called a π 1\pi_1-rationalization if

1. the higher homotopy group π n(X ℚ)\pi_n\big(X_{\mathbb{Q}}\big) are rational vector spaces;

2. η X ℚ\eta^{\mathbb{Q}}_X induces isomorphisms

   1. on the fundamental groups

      π 1(η X ℚ):π 1(X)→∼π 1(X ℚ) \pi_1\big(\eta^{\mathbb{Q}}_{X}\big) \;\colon\; \pi_1(X) \xrightarrow{\;\; \sim \;\;} \pi_1 \big( X_{\mathbb{Q}} \big)

   2. on the the rationalization of all higher homotopy groups π n\pi_n for n≥2n \geq 2:

      π n(η X ℚ)⊗ ℤℚ:π 1(X)⊗ ℤℚ→∼π 1(X ℚ). \pi_n\big(\eta^{\mathbb{Q}}_{X}\big) \otimes_{\mathbb{Z}} \mathbb{Q} \;\colon\; \pi_1(X) \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\;\; \sim \;\;} \pi_1 \big( X_{\mathbb{Q}} \big) \,.

Proposition

(fundamental theorem of dg-algebraic rational homotopy theory)

The derived adjunction

Ho((DiffGradedCommAlgebras k ≥0) proj op)⊥⟶ℝexp⟵𝕃Ω PLdR •Ho(HoSimplicialSets 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( HoSimplicialSets_{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 (2) the derived adjunction unit is rationalization

  Ho(SimplicialSets Qu) ≥1,nil fin ℚ ⟶ Ho(SimplicialSets Qu) ≥1,nil ℚ,fin ℚ X ↦ ℝexp∘Ω 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) }

  X⟶η X derrationalizationℝexp∘Ω 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 ≥1≃⟶ℝexp⟵𝕃Ω PLdR •Ho(HoSimplicialSets Qu) ≥1,nil ℚ,fin ℚ 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( HoSimplicialSets_{Qu} \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil}

Proof

This is effectively a restatement of a result that appears below proposition 15.8 in HalperinThomas and is reproduced in some repackaged form as theorem 2.2 of He06. We recall the model category-theoretic context that allows to rephrase this result in the above form.

Let C={a→c←b}C = \{a \to c \leftarrow b\} be the pullback diagram category.

The homotopy limit functor is the right derived functor ℝlim C\mathbb{R} lim_C for the Quillen adjunction (described in detail at homotopy Kan extension)

[C,sSet] inj⊥⟶lim C⟵constsSet. [C,sSet]_{inj} \underoverset {\underset{lim_C}{\longrightarrow}} {\overset{const}{\longleftarrow}} {\;\;\; \bot \;\;\;} sSet \,.

At model structure on functors it is discussed that composition with the Quillen pair Ω •⊣K\Omega^\bullet \dashv K induces a Quillen adjunction

([C,Ω •]⊣[C,K]):[C,dgAlg op]⊥⟶[C,K]⟵[C,Ω •][C,sSet]. ([C,\Omega^\bullet] \dashv [C,K]) \;\colon\; [C, dgAlg^{op}] \underoverset {\underset{[C,K]}{\longrightarrow}} {\overset{[C,\Omega^\bullet]}{\longleftarrow}} {\;\; \bot \;\;} [C,sSet] \,.

We need to show that for every fibrant and cofibrant pullback diagram F∈[C,sSet]F \in [C,sSet] there exists a weak equivalence

Ω •∘lim CF≃lim CΩ •(F)^, \Omega^\bullet \circ lim_C F \;\; \simeq \;\; lim_C \widehat{\Omega^\bullet(F)} \,,

here Ω •(F)^\widehat{\Omega^\bullet(F)} is a fibrant replacement of Ω •(F)\Omega^\bullet(F) in dgAlg opdgAlg^{op}.

Now, every object f∈[C,sSet] injf \in [C,sSet]_{inj} is cofibrant, and it is fibrant if all three objects F(a)F(a), F(b)F(b) and F(c)F(c) are fibrant and one of the two morphisms is a fibration. We may assume without restriction of generality that it is the morphism F(a)→F(c)F(a) \to F(c) that is a fibration. So we assume that F(a),F(b)F(a), F(b) and F(c)F(c) are three Kan complexes and that F(a)→F(b)F(a) \to F(b) is a Kan fibration. Then lim Clim_C sends FF to the ordinary pullback lim CF=F(a)× F(c)F(b)lim_C F = F(a) \times_{F(c)} F(b) in sSetsSet, and so the left hand side of the above equivalence is

Ω •(F(a)× F(c)F(b)). \Omega^\bullet(F(a) \times_{F(c)} F(b)) \,.

Recall that the Sullivan algebras are the cofibrant objects in dgAlgdgAlg, hence the fibrant objects of dgAlg opdgAlg^{op}. Therefore a fibrant replacement of Ω •(F)\Omega^\bullet(F) may be obtained by

• first choosing a Sullivan model (∧ •V,d V)→≃Ω •(c)(\wedge^\bullet V, d_V) \stackrel{\simeq}{\to} \Omega^\bullet(c)

• then choosing factorizations in dgAlgdgAlg of the composites of this with Ω •(F(c))→Ω •(F(a))\Omega^\bullet(F(c)) \to \Omega^\bullet(F(a)) and Ω •(F(c))→Ω •(F(b))\Omega^\bullet(F(c)) \to \Omega^\bullet(F(b)) into cofibrations follows by weak equivalences.

The result is a diagram

(∧ •U *,d U) ← (∧ •V *,d V) ↪ (∧ •W *,d W) ↓ ≃ ↓ ≃ ↓ ≃ Ω •(F(a)) ← Ω •(F(c)) → Ω •(F(b)) \array{ (\wedge^\bullet U^*, d_U) &\leftarrow& (\wedge^\bullet V^*, d_V) &\hookrightarrow& (\wedge^\bullet W^* , d_W) \\ \downarrow^{\simeq} && \downarrow^{\simeq} && \downarrow^{\simeq} \\ \Omega^\bullet(F(a)) &\stackrel{}{\leftarrow}& \Omega^\bullet(F(c)) &\stackrel{}{\to}& \Omega^\bullet(F(b)) }

that in dgAlg opdgAlg^{op} exhibits a fibrant replacement of Ω •(F)\Omega^\bullet(F). The limit over that in dgAlg opdgAlg^{op} is the colimit

(∧ •U *,d U)⊗ (∧ •V *,d V)(∧ •W *,d W) (\wedge^\bullet U^* , d_U) \otimes_{(\wedge^\bullet V^* , d_V)} (\wedge^\bullet W^* , d_W)

in dgAlgdgAlg. So the statement to be proven is that there exists a weak equivalence

(∧ •U *,d U)⊗ (∧ •V *,d V)(∧ •W *,d W)≃Ω •(F(a)× F(c)F(b)). (\wedge^\bullet U^* , d_U) \otimes_{(\wedge^\bullet V^* , d_V)} (\wedge^\bullet W^* , d_W) \simeq \Omega^\bullet(F(a) \times_{F(c)} F(b)) \,.

This is precisely the statement of that quoted result He, theorem 2.2.

Corollary

Rationalization preserves homotopy pullbacks of objects of finite type.

Proof

The theory of Sullivan models asserts that rationalization of a space XX (a simplicial set XX) is the derived unit of the derived adjunction (Ω •⊣K)(\Omega^\bullet \dashv K), namely that the rationalization is modeled by KK applied to a Sullivan model (∧ •V *,d)(\wedge^\bullet V^*, d) for Ω •(X)\Omega^\bullet(X).

X→KΩ •(X)←≃KΩ •(X)^:=K(∧ •V *,d V). X \to K \Omega^\bullet(X) \stackrel{\simeq}{\leftarrow} K \widehat {\Omega^\bullet(X)} := K (\wedge^\bullet V^* , d_V) \,.

Being a Quillen right adjoint, the right derived functor of KK of course preserves homotopy limits. Hence the composite K∘Ω •(−)^K \circ \widehat{\Omega^\bullet(-)} preserves homotopy pullbacks between objects of finite type.