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\begin{document}

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\section*{rationalization}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{rational_homotopy_theory}{}\paragraph*{{Rational homotopy theory}}\label{rational_homotopy_theory}

[[!include differential graded objects - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{rationalization_of_a_single_space}{Rationalization of a single space}\dotfill \pageref*{rationalization_of_a_single_space} \linebreak
\noindent\hyperlink{rationalization_as_a_localization_of_}{Rationalization as a localization of $Top$/$\infty Grpd$}\dotfill \pageref*{rationalization_as_a_localization_of_} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{preservation_of_homotopy_pullbacks}{Preservation of homotopy pullbacks}\dotfill \pageref*{preservation_of_homotopy_pullbacks} \linebreak
\noindent\hyperlink{RationalizationOfSpectra}{Rationalization of spectra}\dotfill \pageref*{RationalizationOfSpectra} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[rational homotopy theory]] one considers [[topological spaces]] $X$ only up to maps that induce [[isomorphisms]] on \emph{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 \textbf{rationalization}.

Similarly one may consider ``real-ification'' by considering $\pi_\bullet(X) \otimes_{\mathbb{Z}} \mathbb{R}$, etc.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{rationalization_of_a_single_space}{}\subsubsection*{{Rationalization of a single space}}\label{rationalization_of_a_single_space}

A \textbf{rationalization} of a [[simply connected space|simply connected]] [[topological space]] $X$ is a continuous map $\phi : X \to Y$ where

\begin{itemize}%
\item $Y$ is a [[simply connected space|simply connected]] [[rational space]];


\item $\phi$ induces an [[isomorphism]] on rationalized [[homotopy group]]s:

\begin{displaymath}
\pi_\bullet(\phi)\otimes \mathbb{Q} :
  \pi_\bullet(X) \otimes \mathbb{Q}
  \stackrel{\simeq}{\to} 
  \pi_\bullet(Y) \otimes \mathbb{Q}
\end{displaymath}
or equivalently if $\phi$ induces an isomorphism on rational [[homology]] groups

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


\end{itemize}
\hypertarget{rationalization_as_a_localization_of_}{}\subsubsection*{{Rationalization as a localization of $Top$/$\infty Grpd$}}\label{rationalization_as_a_localization_of_}

In [[rational homotopy theory]] one considers the [[Quillen adjunction]]

\begin{displaymath}
(\Omega^\bullet
   \dashv K)
  :
  dgAlg_{\mathbb{Q}} 
   \stackrel{\overset{\Omega^\bullet}{\leftarrow}}{\underset{K}{\to}}
  sSet
\end{displaymath}
between the [[model structure on dg-algebras]] and the standard [[model structure on simplicial sets]], where $\Omega^\bullet$ is forming [[Sullivan differential forms]]:

\begin{displaymath}
\Omega^\bullet(X) = Hom_{sSet}(X, \Omega^\bullet_{pl}(\Delta^\bullet_{Diff}))
  \,.
\end{displaymath}
Intrinsically this should model something like the (partially) left exact [[localization of an (∞,1)-category]] of [[∞Grpd]] at those morphisms that are [[rational homotopy equivalence]]s.

\begin{displaymath}
\infty Grpd_{ratio} \stackrel{\leftarrow}{\hookrightarrow} 
  \infty Grpd
  \,.
\end{displaymath}
Below we review classical results that says that the left [[adjoint (infinity,1)-functor]] here indeed preserves at least [[homotopy pullback]]s.

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]].

\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{preservation_of_homotopy_pullbacks}{}\subsubsection*{{Preservation of homotopy pullbacks}}\label{preservation_of_homotopy_pullbacks}

\begin{theorem}
\label{}\hypertarget{}{}
The left [[derived functor]] of the [[Quillen functor|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]]

\begin{displaymath}
(\Omega^\bullet \dashv K)
   :
  (dgAlg_\mathbb{Q}^{op})^\circ
  \stackrel{\overset{}{\leftarrow}}{\underset{}{\to}}
  \infty Grpd
\end{displaymath}
the left adjoint preserves [[limit in a quasi-category|(∞,1)-categorical pullbacks]] of objects of finite type.

\end{theorem}
\begin{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 \href{http://www.math.uic.edu/~bshipley/hess_ratlhtpy.pdf}{He06}. We recall the [[model category|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]])

\begin{displaymath}
[C,sSet]_{inj} \stackrel{\overset{const}{\leftarrow}}{\underset{lim_C}{\to}}
  sSet
  \,.
\end{displaymath}
At [[model structure on functors]] it is discussed that composition with the Quillen pair $\Omega^\bullet \dashv K$ induces a Quillen adjunction

\begin{displaymath}
([C,\Omega^bullet] \dashv [C,K])
  :
  [C, dgAlg^{op}]
  \stackrel{\overset{[C,\Omega^\bullet]}{\leftarrow}}{\underset{[C,K]}{\to}}
  [C,sSet]
  \,.
\end{displaymath}
We need to show that for every fibrant and cofibrant pullback diagram $F \in [C,sSet]$ there exists a weak equivalence

\begin{displaymath}
\Omega^\bullet \circ lim_C F
  \;\;
  \simeq
  \;\;
  lim_C   \widehat{\Omega^\bullet(F)}
  \,,
\end{displaymath}
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 complex]]es 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

\begin{displaymath}
\Omega^\bullet(F(a) \times_{F(c)} F(b))
  \,.
\end{displaymath}
Recall that the [[Sullivan algebra]]s 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

\begin{itemize}%
\item first choosing a [[Sullivan model]] $(\wedge^\bullet V, d_V) \stackrel{\simeq}{\to} \Omega^\bullet(c)$


\item 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.



\end{itemize}
The result is a diagram

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

\begin{displaymath}
(\wedge^\bullet U^* , d_U)
  \otimes_{(\wedge^\bullet V^* , d_V)}
  (\wedge^\bullet W^* , d_W)
\end{displaymath}
in $dgAlg$. So the statement to be proven is that there exists a weak equivalence

\begin{displaymath}
(\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))
  \,.
\end{displaymath}
This is precisely the statement of that quoted result \href{http://www.math.uic.edu/~bshipley/hess_ratlhtpy.pdf}{He, theorem 2.2}.

\end{proof}
\begin{quote}%
check the following


\end{quote}
\begin{cor}
\label{}\hypertarget{}{}
Rationalization preserves homotopy pullbacks of objects of finite type.

\end{cor}
\begin{proof}
The theory of [[Sullivan model]]s 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 $\Omega^\bullet(X)$.

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

\end{proof}
\hypertarget{RationalizationOfSpectra}{}\subsubsection*{{Rationalization of spectra}}\label{RationalizationOfSpectra}

On [[spectra]], rationalization is a [[smashing localization]], given by [[smash product]] with the [[Eilenberg-MacLane spectrum]] $H \mathbb{Q}$. (e.g. \hyperlink{Bauer11}{Bauer 11, example 1.7 (4)}).

For more see at \emph{[[rational stable homotopy theory]]}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[p-localization]], [[p-completion]]


\item [[fracture square]]


\item [[rational homotopy theory]]


\item [[rational equivariant stable homotopy theory]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Around definition 1.4 in

\begin{itemize}%
\item [[Kathryn Hess]], \emph{Rational homotopy theory: a brief introduction} (\href{http://arxiv.org/abs/math.AT/0604626}{arXiv:math/0604626})


\item [[Tilman Bauer]], \emph{Bousfield localization and the Hasse square} (2011) (\href{http://math.mit.edu/conferences/talbot/2007/tmfproc/Chapter09/bauer.pdf}{pdf})



\end{itemize}
[[!redirects rationalizations]]

[[!redirects rationalisation]] [[!redirects rationalisations]]



\end{document}