nLab
rationalization

Contents

Idea

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.

Definition

Rationalization of a single space

A rationalization of a simply connected topological space XX is a continuous map ϕ:XY\phi : 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} : \pi_\bullet(X) \otimes \mathbb{Q} \stackrel{\simeq}{\to} \pi_\bullet(Y) \otimes \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}) : H_\bullet(X,\mathbb{Q}) \stackrel{\simeq}{\to} H_\bullet(Y,\mathbb{Q}) \,.

Rationalization as a localization of TopTop/Grpd\infty Grpd

In rational homotopy theory one considers the Quillen adjunction

(Ω K):dgAlg KΩ sSet (\Omega^\bullet \dashv K) : 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})) \,.

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.

Properties

Preservation of homotopy pullbacks

Theorem

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.

Proof

This is effectively a restatement of a result that appears effectively 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} \stackrel{\overset{const}{\leftarrow}}{\underset{lim_C}{\to}} 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 bullet][C,K]):[C,dgAlg op][C,K][C,Ω ][C,sSet]. ([C,\Omega^bullet] \dashv [C,K]) : [C, dgAlg^{op}] \stackrel{\overset{[C,\Omega^\bullet]}{\leftarrow}}{\underset{[C,K]}{\to}} [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}.

Every object f[C,sSet] injf \in [C,sSet]_{inj} is cofibrant. 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. Let us 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

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).

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.

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.

References

Around definition 1.4 in

Revised on February 22, 2017 11:08:39 by Urs Schreiber (147.231.89.7)