nLab
rationalization

Idea

In rational homotopy theory one considers topological spaces only up to maps that induce isomorphisms on rationalized homotopy groups.

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

Definition

Rationalization of a single space

A rationalization of a simply connected topological space $X$ is a continuous map $\varphi :X\to Y$ where

• $Y$ is a simply connected rational space;

• $\varphi$ induces an isomorphism on rationalized homotopy groups:

  $${\pi }_{•}(\varphi )\otimes \mathbb{Q}:{\pi }_{•}(X)\otimes \mathbb{Q}\stackrel{\simeq }{\to }{\pi }_{•}(Y)\otimes \mathbb{Q}$$\pi_\bullet(\phi)\otimes \mathbb{Q} : \pi_\bullet(X) \otimes \mathbb{Q} \stackrel{\simeq}{\to} \pi_\bullet(Y) \otimes \mathbb{Q}

  or equivalently if $\varphi$ induces an isomorphism on rational homology groups

  $$H_{•}(\varphi ,\mathbb{Q}):H_{•}(X,\mathbb{Q})\stackrel{\simeq }{\to }{H}_{•}(Y,\mathbb{Q}).$$H_\bullet(\phi,\mathbb{Q}) : H_\bullet(X,\mathbb{Q}) \stackrel{\simeq}{\to} H_\bullet(Y,\mathbb{Q}) \,.

Rationalization as a localization of $\mathrm{Top}$/$\mathrm{\infty }\mathrm{Grpd}$

In rational homotopy theory one considers the Quillen adjunction

$$({\Omega }^{•}⊣K):{\mathrm{dgAlg}}_{\mathbb{Q}}\stackrel{\stackrel{{\Omega }^{•}}{\leftarrow }}{\underset{K}{\to }}\mathrm{sSet}$$(\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 ${\Omega }^{•}$ is forming Sullivan differential forms?:

$${\Omega }^{•}(X)={\mathrm{Hom}}_{\mathrm{sSet}}(X,{\Omega }_{\mathrm{pl}}^{•}({\Delta }_{\mathrm{Diff}}^{•})).$$\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.

$$\mathrm{\infty }{\mathrm{Grpd}}_{\mathrm{ratio}}\overleftarrow{\hookrightarrow }\mathrm{\infty }\mathrm{Grpd}.$$\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

Theorem

The left derived functor of the Quillen left adjoint ${\Omega }^{•}:\mathrm{sSet}\to {\mathrm{dgAlg}}_{\mathbb{Q}}$ 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

$$({\Omega }^{•}⊣K):({\mathrm{dgAlg}}_{\mathbb{Q}}^{\mathrm{op}}{)}^{\circ }\overleftarrow{\underset{}{\to }}\mathrm{\infty }\mathrm{Grpd}$$(\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 $C=\{a\to c\leftarrow b\}$ be the pullback diagram category.

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

$$[C,\mathrm{sSet}{]}_{\mathrm{inj}}\stackrel{\stackrel{\mathrm{const}}{\leftarrow }}{\underset{{\lim }_C}{\to }}\mathrm{sSet}.$$[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 ${\Omega }^{•}⊣K$ induces a Quillen adjunction

$$([C,{\Omega }^{\mathrm{bullet}}]⊣[C,K]):[C,{\mathrm{dgAlg}}^{\mathrm{op}}]\stackrel{\stackrel{[C,{\Omega }^{•}]}{\leftarrow }}{\underset{[C,K]}{\to }}[C,\mathrm{sSet}].$$([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 $F\in [C,\mathrm{sSet}]$ there exists a weak equivalence

$${\Omega }^{•}\circ {\lim }_{C}F\simeq {\lim }_C\widehat{{\Omega }^{•}(F)},$$\Omega^\bullet \circ lim_C F \;\; \simeq \;\; lim_C \widehat{\Omega^\bullet(F)} \,,

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

Every object $f\in [C,\mathrm{sSet}{]}_{\mathrm{inj}}$ is cofibrant. It is fibrant if all three objects $F(a)$, $F(b)$ and $F(c)$ are fibrant and one of the two morphisms is a fibration. Let us assume without restriction of generality that it is the morphism $F(a)\to F(c)$ that is a fibration. So we assume that $F(a),F(b)$ and $F(c)$ are three Kan complexes and that $F(a)\to F(b)$ is a Kan fibration. Then ${\lim }_C$ sends $F$ to the ordinary pullback ${\lim }_{C}F=F(a){\times }_{F(c)}F(b)$ in $\mathrm{sSet}$, and so the left hand side of the above equivalence is

$${\Omega }^{•}(F(a){\times }_{F(c)}F(b)).$$\Omega^\bullet(F(a) \times_{F(c)} F(b)) \,.

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

• first choosing a Sullivan model $({\wedge }^{•}V,d_V)\stackrel{\simeq }{\to }{\Omega }^{•}(c)$

• then choosing factorizations in $\mathrm{dgAlg}$ of the composites of this with ${\Omega }^{•}(F(c))\to {\Omega }^{•}(F(a))$ and ${\Omega }^{•}(F(c))\to {\Omega }^{•}(F(b))$ into cofibrations follows by weak equivalences.

The result is a diagram

$$\begin{array}{ccccc}({\wedge }^{•}{U}^{*},d_U)& \leftarrow & ({\wedge }^{•}{V}^{*},d_V)& \hookrightarrow & ({\wedge }^{•}{W}^{*},d_W)\\ {\downarrow }^{\simeq }& & {\downarrow }^{\simeq }& & {\downarrow }^{\simeq }\\ {\Omega }^{•}(F(a))& \stackrel{}{\leftarrow }& {\Omega }^{•}(F(c))& \stackrel{}{\to }& {\Omega }^{•}(F(b))\end{array}$$\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 ${\mathrm{dgAlg}}^{\mathrm{op}}$ exhibits a fibrant replacement of ${\Omega }^{•}(F)$. The limit over that in ${\mathrm{dgAlg}}^{\mathrm{op}}$ is the colimit

$$({\wedge }^{•}{U}^{*},d_U){\otimes }_{({\wedge }^{•}{V}^{*},d_V)}({\wedge }^{•}{W}^{*},d_W)$$(\wedge^\bullet U^* , d_U) \otimes_{(\wedge^\bullet V^* , d_V)} (\wedge^\bullet W^* , d_W)

in $\mathrm{dgAlg}$. So the statement to be proven is that there exists a weak equivalence

$$({\wedge }^{•}{U}^{*},d_U){\otimes }_{({\wedge }^{•}{V}^{*},d_V)}({\wedge }^{•}{W}^{*},d_W)\simeq {\Omega }^{•}(F(a){\times }_{F(c)}F(b)).$$(\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 $X$ (a simplicial set $X$) is the derived unit of the derived adjunction $({\Omega }^{•}⊣K)$, namely that the rationalization is modeled by $K$ applied to a Sullivan model $({\wedge }^{•}{V}^{*},d)$ for ${\Omega }^{•}(X)$.

$$X\to K{\Omega }^{•}(X)\stackrel{\simeq }{\leftarrow }K\widehat{{\Omega }^{•}(X)}:=K({\wedge }^{•}{V}^{*},d_V).$$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 $K$ of course preserves homotopy limits. Hence the composite $K\circ \widehat{{\Omega }^{•}(-)}$ preserves homotopy pullbacks between objects of finite type.

References

Around definition 1.4 in

• Kathryn Hess, Rational homotopy theory: a brief introduction (arXiv:math/0604626)