nLab rationalization

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 homotopy type

We discuss here what it means for a map of homotopy types to exhibit the rationalization of its domain.

A widely appreciated construction applies in the special case that the domain is a simply connected homotopy type (in fact, more generally in the case that it is a nilpotent homotopy type). A slight enhancement of this construction, which is generally much less widely considered, applies to all connected homotopy types: this forms the rationalization of the universal cover (which is of course simply connected) but retains on this the information of the $\infty$-actions of the fundamental groups by deck transformations (see also at Borel-equivariant rational homotopy theory):

1. Rationalization of simply-connected homotopy types

2. $\pi_1$-Rationalization of connected homotopy types

Rationalization of simply-connected spaces

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

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

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

(notice here that $\mathbb{Z}$ is a solid ring, in that $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q}\,\simeq\, \mathbb{Q}$, e.g. here)

Proposition

(rationalization via $\mathbb{Q}$-completion of simplicial loop group)
The rationalization of a simply connected homotopy type is represented (say via the classical model structure on simplicial sets) by a reduced simplicial set $S \,\in\, sSet_\ast$ is given by

$$S_{\mathbb{Q}} \;\simeq\; \overline{W}(G S)_{\widehat{\mathbb{Q}}} \;\;\; \in sSet \xrightarrow{ L^W } Grp_\infty \,,$$

where:

1. $G \;\colon\; sSet_\ast \xrightarrow{\;} sGrp$ is the simplicial loop space-functor to simplicial groups;

2. $(-)_{\widehat{\mathbb{Q}}} \;\colon\; sGrp \to sGrp$ is degreewise the $\mathbb{Q}$-completion (hence Malcev completion) applied to the component group of a simplicial group;

3. $\overline{W} \;\colon\; sGrp \xrightarrow{\;} sSet$ is the simplicial delooping-operation.

(Bousfield & Kan 1971, §3, Bousfield & Kan 1972, IV Prop. 4.1 (p. 109), see also Rivera, Wierstra & Zeinalian 2021, p. 7)

$\pi_1$-Rationalization of connected spaces

For connected homotopy types $X$ which are not necessarily simply connected, consider their universal cover $\widehat{X}$, which sits in a homotopy fiber sequence

(1)$$\array{ \widehat{X} &\longrightarrow& X \\ \big\downarrow && \big\downarrow \mathrlap{ {}^{ \eta^{[-]_1}_X } } \\ \ast &\longrightarrow& B \pi_1(X) }$$

over the delooping/classifying space of the fundamental group (via the 1-truncation unit).

(Notice that this fiber sequence exhibits the $\infty$-action of $\pi_1(X)$ on $\widehat{X}$.)

Here the universal cover $\widehat{X}$ is simply connected (hence in particular nilpotent), so that the above notion of rationalization applies to this fiber:

Definition

($\pi_1$-rationalization)
A map $\eta^{\mathbb{Q}}_X \;\colon\; X \longrightarrow X_{\mathbb{Q}}$ of connected homotopy types is called a $\pi_1$-rationalization if

1. the higher homotopy group $\pi_n\big(X_{\mathbb{Q}}\big)$ are rational vector spaces;

2. $\eta^{\mathbb{Q}}_X$ induces isomorphisms

   1. on the fundamental groups

      $\pi_1\big(\eta^{\mathbb{Q}}_{X}\big) \;\colon\; \pi_1(X) \xrightarrow{\;\; \sim \;\;} \pi_1 \big( X_{\mathbb{Q}} \big)$

   2. on the the rationalization of all higher homotopy groups $\pi_n$ for $n \geq 2$:

      $\pi_n\big(\eta^{\mathbb{Q}}_{X}\big) \otimes_{\mathbb{Z}} \mathbb{Q} \;\colon\; \pi_1(X) \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\;\; \sim \;\;} \pi_1 \big( X_{\mathbb{Q}} \big) \,.$

In generalization of Prop. :

Proposition

The rationalization (Def. ) of a connected homotopy type is represented (say via the classical model structure on simplicial sets) by a reduced simplicial set $S \,\in\, sSet_\ast$ is given by

$$S_{\mathbb{Q}} \;\simeq\; \overline{W}(G S)_{\widehat{\mathbb{Q}/\pi}} \;\;\; \in sSet \xrightarrow{ L^W } Grp_\infty \,,$$

where:

1. $G \;\colon\; sSet_\ast \xrightarrow{\;} sGrp$ is the simplicial loop space-functor to simplicial groups;

2. $(-)_{\widehat{\mathbb{Q}/\pi}}$ is in degree $n$ the fiberwise rationalization of of the short exact sequences of homotopy groups (obtained in the present case from the long exact sequence of homotopy groups of the fiber sequence (1))

   $$1 \to \pi_n \big( \widehat{X} \big) \longrightarrow \pi_n \big( X \big) \longrightarrow \pi_n \big( B \pi_1(X) \big) \to 1$$

   (in the present case this is trivial in degree 1 and is the plain rationalization in higher degrees, but this formulation makes manifest that both cases are functorially compatible);

3. $\overline{W} \;\colon\; sGrp \xrightarrow{\;} sSet$ is the simplicial delooping-operation.

(Bousfield & Kan 1971, §3, p. 1008, see also Rivera, Wierstra & Zeinalian 2021, p. 8 and Ivanov 2021).

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 cohomology 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.

In homotopy type theory

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

Properties

Rationalization via PL de Rham theory

Definition

(nilpotent and finite rational homotopy types)

Write

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

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

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

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

Similarly, write

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

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 $n \in \mathbb{N}$.

(Bousfield-Gugenheim 76, 9.2)

Proposition

(fundamental theorem of dg-algebraic rational homotopy theory)

The derived adjunction

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

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 $X$ (2) 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:

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

(Bousfield-Gugenheim 76, Theorems 9.4 & 11.2)

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.

Related concepts

• p-localization, p-completion

• equivariant rationalization

• fracture square

• rational homotopy theory

  • Borel-equivariant rational homotopy theory

• fundamental theorem of dg-algebraic rational homotopy theory

• rational equivariant homotopy theory

• rational equivariant stable homotopy theory

References

Classical accounts:

• Aldridge Bousfield, Daniel Kan, Localization and completion in homotopy theory, Bull. Amer. Math. Soc. 77 6 (1971) 1006-1010 [doi:10.1090/S0002-9904-1971-12837-9, pdf]

• Aldridge Bousfield, Daniel Kan, p. 133-140 in: Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics Vol. 304, Springer 1972 (doi:10.1007/978-3-540-38117-4)

• Aldridge Bousfield, Victor Gugenheim, Def. 11.1 in: On PL deRham theory and rational homotopy type, Memoirs of the AMS, vol. 179 (1976) (ams:memo-8-179)

Review:

• Kathryn Hess, Def. 1.7 in: Rational homotopy theory: a brief introduction, contribution to Summer School on Interactions between Homotopy Theory and Algebra, University of Chicago, July 26-August 6, 2004, Chicago (arXiv:math.AT/0604626), chapter in Luchezar Lavramov, Dan Christensen, William Dwyer, Michael Mandell, Brooke Shipley (eds.), Interactions between Homotopy Theory and Algebra, Contemporary Mathematics 436, AMS 2007 (doi:10.1090/conm/436)

• Tilman Bauer, Bousfield localization and the Hasse square (2011) (pdf, pdf), chapter 6 in: Christopher Douglas, John Francis, André Henriques, Michael Hill (eds.), Topological Modular Forms, Mathematical Surveys and Monographs Volume 201, AMS 2014 (ISBN:978-1-4704-1884-7)

• Tyler Lawson, Example 8.12 in: An introduction to Bousfield localization (arXiv:2002.03888) in: Andrew J. Blumberg, Teena Gerhardt, Michael A. Hill (eds.), Stable categories and structured ring spectra MSRI Book Series, Cambridge University Press

Review and further developments:

• Manuel Rivera, Felix Wierstra, Mahmoud Zeinalian, Rational homotopy equivalences and singular chains, Algebr. Geom. Topol. 21 (2021) 1535-1552 [arXiv:1906.03655, doi:10.2140/agt.2021.21.1535]

• Sergei O. Ivanov, An overview of rationalization theories of non-simply connected spaces and non-nilpotent groups [arXiv:2111.10694]