nLab rationalization




In rational homotopy theory one considers topological spaces XX only up to maps that induce isomorphisms on rationalized homotopy groups π (X) \pi_\bullet(X) \otimes_{\mathbb{Z}} \mathbb{Q} (as opposed to genuine weak homotopy equivalences, which are those maps that induce isomorphism on the genuine homotopy groups.)

Every simply connected space is in this sense equivalent to a rational space: this is its rationalization.

Similarly one may consider “real-ification” by considering π (X) \pi_\bullet(X) \otimes_{\mathbb{Z}} \mathbb{R}, etc.


Rationalization of a single space

A rationalization of a simply connected topological space XX is a continuous function ϕ:XY\phi \colon X \to Y, where

  • YY is a simply connected rational space;

  • ϕ\phi induces an isomorphism on rationalized homotopy groups:

    π (ϕ):π (X)π (Y) \pi_\bullet(\phi)\otimes \mathbb{Q} \;\colon\; \pi_\bullet(X) \otimes \mathbb{Q} \stackrel{\simeq}{\longrightarrow} \pi_\bullet(Y) \otimes \mathbb{Q}

    or equivalently if ϕ\phi induces an isomorphism on rational cohomology groups

    H (ϕ,):H (X,)H (Y,). H^\bullet(\phi,\mathbb{Q}) \;\colon\; H^\bullet(X,\mathbb{Q}) \stackrel{\simeq}{\longrightarrow} H^\bullet(Y,\mathbb{Q}) \,.

    or equivalently if ϕ\phi induces an isomorphism on rational homology groups

    H (ϕ,):H (X,)H (Y,). H_\bullet(\phi,\mathbb{Q}) \;\colon\; H_\bullet(X,\mathbb{Q}) \stackrel{\simeq}{\longrightarrow} H_\bullet(Y,\mathbb{Q}) \,.

(Bousfield-Kan 72, p. 133-140, Bousfield-Gugenheim 76, 11.1, Hess 06, Def. 1.4 with Def. 1.7)

(notice here that \mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q}\,\simeq\, \mathbb{Q}, e.g. here)

Rationalization as a localization of TopTop/Grpd\infty Grpd

In rational homotopy theory one considers the PL de Rham Quillen adjunction

(Ω K):dgAlg KΩ sSet (\Omega^\bullet \dashv K) \;\colon\; dgAlg_{\mathbb{Q}} \stackrel{\overset{\Omega^\bullet}{\leftarrow}}{\underset{K}{\to}} sSet

between the model structure on dg-algebras and the standard model structure on simplicial sets, where Ω \Omega^\bullet is forming Sullivan differential forms:

Ω (X)=Hom sSet(X,Ω pl (Δ Diff )). \Omega^\bullet(X) = Hom_{sSet}(X, \Omega^\bullet_{pl}(\Delta^\bullet_{Diff})) \,.

The fundamental theorem of dg-algebraic rational homotopy theory says that on nilpotent spaces with finite type rational chomology? this induces an equivalence of homotopy categories.

Intrinsically this should model something like the (partially) left exact localization of an (∞,1)-category of ∞Grpd at those morphisms that are rational homotopy equivalences.

Grpd ratioGrpd. \infty Grpd_{ratio} \stackrel{\leftarrow}{\hookrightarrow} \infty Grpd \,.

Below we review classical results that says that the left adjoint (infinity,1)-functor here indeed preserves at least homotopy pullbacks.

More generally, a setup by Bertrand Toen serves to provide a more comprehensive description of this situtation: see rational homotopy theory in an (infinity,1)-topos.

In homotopy type theory

In homotopy type theory and synthetic homotopy theory, one considers the rationalization of a pointed 1-connected? type.


Rationalization via PL de Rham theory


(nilpotent and finite rational homotopy types)


(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}.


(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 futher 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)


(fundamental theorem of dg-algebraic rational homotopy theory)

The derived adjunction

Ho((DiffGradedCommAlgebras k 0) proj op)exp𝕃Ω 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 (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(HoSimplicialSets 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( HoSimplicialSets_{Qu} \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil}

(Bousfield-Gugenheim 76, Theorems 9.4 & 11.2)

Preservation of homotopy pullbacks


The left derived functor of the Quillen left adjoint Ω :sSetdgAlg \Omega^\bullet : sSet \to dgAlg_{\mathbb{Q}} preserves homotopy pullbacks of objects of finite type (each rational homotopy group is a finite dimensional vector space over the ground field).

In other words in the induced pair of adjoint (∞,1)-functors

(Ω K):(dgAlg op) Grpd (\Omega^\bullet \dashv K) : (dgAlg_\mathbb{Q}^{op})^\circ \stackrel{\overset{}{\leftarrow}}{\underset{}{\to}} \infty Grpd

the left adjoint preserves (∞,1)-categorical pullbacks of objects of finite type.


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

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

[C,sSet] injlim CconstsSet. [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 CFlim 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.

check the following


Rationalization preserves homotopy pullbacks of objects of finite type.


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).

XKΩ (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 KΩ ()^K \circ \widehat{\Omega^\bullet(-)} preserves homotopy pullbacks between objects of finite type.

Preservation of homotopy fibers

See at rational fibration lemma.

Rationalization of spectra

On spectra, rationalization is a smashing localization, given by smash product with the Eilenberg-MacLane spectrum HH \mathbb{Q}. (e.g. Bauer 11, example 1.7 (4)).

For more see at rational stable homotopy theory.


Last revised on June 17, 2022 at 17:14:22. See the history of this page for a list of all contributions to it.