# nLab rationalization

Contents

### Context

#### Rational homotopy theory

and

rational homotopy theory

# Contents

## Idea

In rational homotopy theory one considers topological spaces $X$ only up to maps that induce isomorphisms on rationalized homotopy groups $\pi_\bullet(X) \otimes_{\mathbb{Z}} \mathbb{Q}$ (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 $\pi_\bullet(X) \otimes_{\mathbb{Z}} \mathbb{R}$, etc.

## Definition

### Rationalization of a single space

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

• $Y$ is a simply connected rational space;

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

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

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

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

### Rationalization as a localization of $Top$/$\infty Grpd$

In rational homotopy theory one considers the Quillen adjunction

$(\Omega^\bullet \dashv K) : dgAlg_{\mathbb{Q}} \stackrel{\overset{\Omega^\bullet}{\leftarrow}}{\underset{K}{\to}} sSet$

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

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

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.

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

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 $\Omega^\bullet : sSet \to dgAlg_{\mathbb{Q}}$ 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

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

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 $\mathbb{R} lim_C$ for the Quillen adjunction (described in detail at homotopy Kan extension)

$[C,sSet]_{inj} \stackrel{\overset{const}{\leftarrow}}{\underset{lim_C}{\to}} sSet \,.$

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

$([C,\Omega^bullet] \dashv [C,K]) : [C, dgAlg^{op}] \stackrel{\overset{[C,\Omega^\bullet]}{\leftarrow}}{\underset{[C,K]}{\to}} [C,sSet] \,.$

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

$\Omega^\bullet \circ lim_C F \;\; \simeq \;\; lim_C \widehat{\Omega^\bullet(F)} \,,$

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

Every object $f \in [C,sSet]_{inj}$ 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 $sSet$, and so the left hand side of the above equivalence is

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

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

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

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

The result is a diagram

$\array{ (\wedge^\bullet U^*, d_U) &\leftarrow& (\wedge^\bullet V^*, d_V) &\hookrightarrow& (\wedge^\bullet W^* , d_W) \\ \downarrow^{\simeq} && \downarrow^{\simeq} && \downarrow^{\simeq} \\ \Omega^\bullet(F(a)) &\stackrel{}{\leftarrow}& \Omega^\bullet(F(c)) &\stackrel{}{\to}& \Omega^\bullet(F(b)) }$

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

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

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

$(\wedge^\bullet U^* , d_U) \otimes_{(\wedge^\bullet V^* , d_V)} (\wedge^\bullet W^* , d_W) \simeq \Omega^\bullet(F(a) \times_{F(c)} F(b)) \,.$

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 $(\Omega^\bullet \dashv K)$, namely that the rationalization is modeled by $K$ applied to a Sullivan model $(\wedge^\bullet V^*, d)$ for $\Omega^\bullet(X)$.

$X \to K \Omega^\bullet(X) \stackrel{\simeq}{\leftarrow} K \widehat {\Omega^\bullet(X)} := K (\wedge^\bullet V^* , d_V) \,.$

Being a Quillen right adjoint, the right derived functor of $K$ of course preserves homotopy limits. Hence the composite $K \circ \widehat{\Omega^\bullet(-)}$ 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 $H \mathbb{Q}$. (e.g. Bauer 11, example 1.7 (4)).

For more see at rational stable homotopy theory.

Around definition 1.4 in