Contents

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# 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 function $\phi \colon X \to Y$, where

• $Y$ is a simply connected rational space;

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

### Rationalization via PL de Rham theory

###### Definition

(nilpotent and finite rational homotopy types)

Write

(1)$Ho \big( SimplicialSets_{Qu} \big)^{fin_{\mathbb{Q}}}_{\geq 1, nil} \overset{ \phantom{AAA} }{\hookrightarrow} Ho \big( SimplicialSets_{Qu} \big)$

for the full subcategory of the classical homotopy category (homotopy category of the classical model structure on simplicial sets) on those homotopy types $X$ which are

• connected: $\pi_0(X) = \ast$

• nilpotent: $\pi_1(X)$ is a nilpotent group

• rational finite type: $dim_{\mathbb{Q}}\big( H^n(X;,\mathbb{Q}) \big) \lt \infty$ for all $n \in \mathbb{N}$.

and

(2)$Ho \big( SimplicialSets_{Qu} \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil} \overset{ \phantom{AAA} }{\hookrightarrow} Ho \big( SimplicialSets_{Qu} \big)$

for the futher full subcategory on those homotopy types that are already rational.

Similarly, write

(3)$Ho \big( DiffGradedCommAlgebras^{\geq 0}_{\mathbb{Q}} \big)_{fin}^{\geq 1} \overset{ \phantom{AAA} }{\hookrightarrow} Ho \big( DiffGradedCommAlgebras^{\geq 0}_{\mathbb{Q}} \big)$

for the full subcategory of the homotopy category of the projective model structure on connective dgc-algebras on those dgc-algebras $A$ which are

• connected: $H^0(A) \simeq \mathbb{Q}$

• finite type: $dim_{\mathbb{Q}}\big( H^n(A) \big) \lt \infty$ for all $n \in \mathbb{N}$.

###### Proposition

(fundamental theorem of dg-algebraic rational homotopy theory)

$Ho \left( \big( DiffGradedCommAlgebras^{\geq 0}_{k} \big)^{op}_{proj} \right) \underoverset { \underset {\;\;\; \mathbb{R} exp \;\;\;} {\longrightarrow} } { \overset {\;\;\; \mathbb{L} \Omega^\bullet_{PLdR}\;\;\;} {\longleftarrow} } {\bot} Ho \big( HoSimplicialSets_{Qu} \big)$

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 $X$ (1) the derived adjunction unit is rationalization

$\array{ Ho \big( SimplicialSets_{Qu} \big)^{fin_{\mathbb{Q}}}_{\geq 1, nil} & \overset{ }{\longrightarrow} & Ho \big( SimplicialSets_{Qu} \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil} \\ X &\mapsto& \mathbb{R}\exp \circ \Omega^\bullet_{PLdR}(X) }$
$X \underoverset {\eta_X^{der}} {rationalization} {\longrightarrow} \mathbb{R}\exp \circ \Omega^\bullet_{PLdR}(X)$
• on the full subcategories of nilpotent and finite rational homotopy types from Def. it restricts to an equivalence of categories:

$Ho \left( \big( DiffGradedCommAlgebras^{\geq 0}_{k} \big)^{op}_{proj} \right)^{\geq 1}_{fin} \underoverset { \underset {\;\;\; \mathbb{R} exp \;\;\;} {\longrightarrow} } { \overset {\;\;\; \mathbb{L} \Omega^\bullet_{PLdR}\;\;\;} {\longleftarrow} } {\simeq} Ho \big( HoSimplicialSets_{Qu} \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil}$

### 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 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)

$[C,sSet]_{inj} \underoverset {\underset{lim_C}{\longrightarrow}} {\overset{const}{\longleftarrow}} {\;\;\; \bot \;\;\;} sSet \,.$

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

$([C,\Omega^\bullet] \dashv [C,K]) \;\colon\; [C, dgAlg^{op}] \underoverset {\underset{[C,K]}{\longrightarrow}} {\overset{[C,\Omega^\bullet]}{\longleftarrow}} {\;\; \bot \;\;} [C,sSet] \,.$

We need to show that for every fibrant and cofibrant pullback diagram $F \in [C,sSet]$ 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 $dgAlg^{op}$.

Now, every object $f \in [C,sSet]_{inj}$ is cofibrant, and 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. We may 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.

### 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 $H \mathbb{Q}$. (e.g. Bauer 11, example 1.7 (4)).

For more see at rational stable homotopy theory.