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\begin{document}

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\section*{localization of a space}

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

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{localization_of_a_space}{}\section*{{Localization of a space}}\label{localization_of_a_space}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{preliminaries}{Preliminaries}\dotfill \pageref*{preliminaries} \linebreak
\noindent\hyperlink{sets_of_primes}{Sets of primes}\dotfill \pageref*{sets_of_primes} \linebreak
\noindent\hyperlink{local_groups}{Local groups}\dotfill \pageref*{local_groups} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{localization_of_spectra}{Localization of spectra}\dotfill \pageref*{localization_of_spectra} \linebreak
\noindent\hyperlink{localization_of_nilpotent_spaces}{Localization of nilpotent spaces}\dotfill \pageref*{localization_of_nilpotent_spaces} \linebreak
\noindent\hyperlink{localization_of_nonnilpotent_spaces}{Localization of non-nilpotent spaces}\dotfill \pageref*{localization_of_nonnilpotent_spaces} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToCompletion}{Relation to formal completion}\dotfill \pageref*{RelationToCompletion} \linebreak
\noindent\hyperlink{basic_properties}{Basic properties}\dotfill \pageref*{basic_properties} \linebreak
\noindent\hyperlink{in_chromatic_homotopy_theory}{In chromatic homotopy theory}\dotfill \pageref*{in_chromatic_homotopy_theory} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

The \textbf{localization of a space} (really: [[homotopy type]], [[∞-groupoid]]) or [[spectrum]] with respect to some [[prime numbers]] is a [[homotopy theory|homotopical]] analogue of the notion of [[localization of a commutative ring]] (or rather of [[localization of a module]]), specifically the ring $\mathbb{Z}$ of [[integers]], with respect to some [[prime numbers]]. In good cases, at least, this localization acts on the [[homotopy groups|homotopy]] and/or [[homology groups]] as algebraic localization. (For spaces, there are multiple inequivalent notions of localization, although all agree on [[nilpotent spaces]].)

Localization in this sense is closely related to [[Bousfield localization]]. The localization of spectra is a [[Bousfield localization of spectra]], while one of the constructions of localization of spaces is a Bousfield localization of model categories. The present notion of localization should not be confused with the [[completion of a space]], which is a different sort of Bousfield localization.

\hypertarget{preliminaries}{}\subsection*{{Preliminaries}}\label{preliminaries}

\hypertarget{sets_of_primes}{}\subsubsection*{{Sets of primes}}\label{sets_of_primes}

For all of this page,

\begin{itemize}%
\item let $T$ be a [[set]] of [[prime numbers]],


\item write $\neg T$ for the set of primes not in $T$.



\end{itemize}
\begin{defn}
\label{}\hypertarget{}{}
Write $\mathbb{Z}_T$ for the [[commutative ring|ring]] of [[integers]] [[localization of a commutative ring|localized]] by inverting all primes in $\neg T$, i.e. the subring of $\mathbb{Q}$ whose denominators are products of primes in $\neg T$.

\end{defn}
\begin{example}
\label{}\hypertarget{}{}
The most important cases are:

\begin{itemize}%
\item $T = \{p\}$ for a prime $p$. In this case, $T$-localization will be \textbf{localization at $p$} or \textbf{$p$-localization}.


\item $T = \neg \{p\}$, the set of all primes except $p$. In this case, $T$-localization will be \textbf{localization away from $p$}.


\item $T = \emptyset$. In this case, $T$-localization will be \textbf{[[rationalization]]}.



\end{itemize}
In these cases:

\begin{itemize}%
\item $\mathbb{Z}_\emptyset = \mathbb{Q}$ is the [[rational numbers]];


\item $\mathbb{Z}_{\neg\{p\}} = \mathbb{Z}[\frac{1}{p}]$;


\item $\mathbb{Z}_{\{p\}} = \mathbb{Z}_{(p)}$ is the integers localized at the [[prime ideal]] $(p)$.



\end{itemize}
\end{example}
\begin{remark}
\label{}\hypertarget{}{}
The analogous theory of the [[completion of a space]] involves the [[cyclic groups]] $\mathbb{Z}/p\mathbb{Z}$ -- written $\mathbb{F}_p$ when regarded as a [[finite field]] -- and/or the [[p-adic integers]] $\mathbb{Z}_p$ instead of $\mathbb{Z}_T$ (e.g. \hyperlink{LurieProper}{Lurie ``Proper morphisms'', section 4}).

For the relation of that to completion see remark \ref{RelationToCompletion} below.

Hence beware the subtle but crucial difference in what a subscript means, depending on which symbol is being subscribed and whether there are parenthesis or not:

\begin{itemize}%
\item $\mathbb{Z}_{(p)}$: inverting all primes except $p$;


\item $\mathbb{F}_p$: quotient by $p$;


\item $\mathbb{Z}_p$: [[formal completion]] at $p$.



\end{itemize}
\end{remark}
\hypertarget{local_groups}{}\subsubsection*{{Local groups}}\label{local_groups}

\begin{defn}
\label{TLocalGroup}\hypertarget{TLocalGroup}{}
A [[group]] $G$ is said to be \textbf{$T$-local} if the $p^{th}$ power map $G\to G$ is a bijection for all $p\in \neg T$.

\end{defn}
\begin{prop}
\label{CharacterizationOfAbelianLocalGroups}\hypertarget{CharacterizationOfAbelianLocalGroups}{}
If $G$ is [[abelian group|abelian]], then this map is a group [[homomorphism]] and is generally written additively as multiplication by $p$. In this case the following are equivalent:

\begin{itemize}%
\item $G$ is $T$-local;


\item $G$ admits a structure of $\mathbb{Z}_T$-[[module]] (necessarily unique);


\item The [[tensor product]] $G\otimes \mathbb{Z}/p\mathbb{Z}$ with the [[cyclic group]] of order $p$ is equal to zero for all $p\in\neg T$.



\end{itemize}
\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
The second characterization in prop. \ref{CharacterizationOfAbelianLocalGroups} implies that $T$-local abelian groups are [[reflective subcategory|reflective]] in [[Ab]]: the reflection is the [[extension of scalars]] functor $(\mathbb{Z}_T \otimes -)$. In fact, $T$-local nonabelian groups are also reflective in [[Grp]], but the construction is less pretty.

\end{prop}
\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\hypertarget{localization_of_spectra}{}\subsubsection*{{Localization of spectra}}\label{localization_of_spectra}

\begin{defn}
\label{TLocalSpectrum}\hypertarget{TLocalSpectrum}{}
A [[spectrum]] $X$ is called \textbf{$T$-local} (or $\mathbb{Z}_T$-local, if there is potential for confusion) if its [[homotopy groups]] are $T$-local abelian groups, def. \ref{TLocalGroup}.

The \textbf{$T$-localization} of a spectrum is its [[reflective sub-(infinity,1)-category|reflection]] into $T$-local spectra.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The $T$-localization, def. \ref{TLocalSpectrum}, may be constructed as the [[Bousfield localization of spectra]] with respect to the [[Moore spectrum]] $S(\mathbb{Z}_T)$ (e.g. \hyperlink{Bauer11}{Bauer 11, Example 1.7}).

It can also be constructed as a [[Bousfield localization of model categories]] where we invert the maps that induce an isomorphism on [[generalized homology]] with [[coefficients]] in $H \mathbb{Z}_T$; these are called \textbf{$\mathbb{Z}_T$-homology isomorphisms}. See at \emph{[[homology localization]]}.

\end{remark}
\hypertarget{localization_of_nilpotent_spaces}{}\subsubsection*{{Localization of nilpotent spaces}}\label{localization_of_nilpotent_spaces}

The presence of the nonabelian group $\pi_1$ makes the theory of localization of unstable spaces more subtle than that of spectra. If spaces are [[simply connected]], of course, then this is not a problem. More generally, it suffices to consider [[simple spaces]]: those where $\pi_1$ is abelian and acts trivially on the higher homotopy groups. Even more generally, it suffices to consider [[nilpotent spaces]], whose definition is more complicated; the reader is encouraged to think about simple or even simply connected spaces.

For a nilpotent space $Z$, the following conditions are equivalent. When they hold, we say that $Z$ is \textbf{$T$-local}. (See \hyperlink{MayPonto}{May-Ponto, Theorem 6.1.1}.)

\begin{enumerate}%
\item Whenever $X\to Y$ induces an isomorphism on homology with $\mathbb{Z}_T$-coefficients, the induced map $[Y,Z] \to [X,Z]$ is a bijection.


\item Each [[homotopy group]] $\pi_n(Z)$ is a $T$-local group.


\item Each [[homology group]] $H_n(Z,\mathbb{Z})$ is a $T$-local group.



\end{enumerate}
For a general nilpotent space $X$, the following properties of a map $\phi:X\to Y$, with $Y$ nilpotent and $T$-local, are equivalent. Such a map exists and is unique up to homotopy, and we call it the \textbf{$T$-localization} of $X$ at $T$. (See \hyperlink{MayPonto}{May-Ponto, Theorem 6.1.2}.)

\begin{enumerate}%
\item The induced map $[Y,Z] \to [X,Z]$ is a bijection for all $T$-local nilpotent spaces $Z$.


\item $\phi$ induces an isomorphism on homology with $\mathbb{Z}_T$-coefficients.


\item The induced map $\pi_n(X) \to \pi_n(Y)$ is an algebraic $T$-localization for all $n$.


\item The induced map $H_n(X,\mathbb{Z}) \to H_n(Y,\mathbb{Z})$ is an algebraic $T$-localization for all $n$.



\end{enumerate}
\hypertarget{localization_of_nonnilpotent_spaces}{}\subsubsection*{{Localization of non-nilpotent spaces}}\label{localization_of_nonnilpotent_spaces}

For non-nilpotent spaces, there are multiple inequivalent definitions of localization (see \hyperlink{MayPonto}{May-Ponto, Remark 19.3.11}). For instance:

\begin{itemize}%
\item One can do a [[Bousfield localization of model categories]] with respect to the class of $\mathbb{Z}_T$-homology isomorphisms. In this case, the ``$T$-local spaces'' are those $Z$ such that $[Y,Z] \to [X,Z]$ is an isomorphism for all $\mathbb{Z}_T$-homology isomorphisms $X\to Y$ (the first characterization of $T$-local nilpotent spaces above).


\item One can construct a [[totalization]] of the [[cosimplicial object]] induced by the ``free simplicial $\mathbb{Z}_T$-module'' monad. This construction is due to \hyperlink{BousfieldKan72}{Bousfield-Kan}.


\item One can do a [[Bousfield localization of model categories]] with respect to the maps $p:S^1\to S^1$ for all $p\in \neg T$. This construction is due to \hyperlink{CasacubertaPeschke}{Casacuberta-Peschke}. In this case, the ``$T$-local spaces'' are those such that $\pi_1(X)$ is a $T$-local group and acts ``$T$-[[p-local module|locally]]'' on each $\pi_n(X)$.



\end{itemize}
One can of course state the other two characterizations of ``$T$-local space'' from the nilpotent case even in the non-nilpotent case, as is done by \hyperlink{Sullivan70}{Sullivan}, but these characterizations are no longer equivalent to the first one, and it is not clear whether corresponding ``localizations'' exist.

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

\hypertarget{RelationToCompletion}{}\subsubsection*{{Relation to formal completion}}\label{RelationToCompletion}

\begin{remark}
\label{RelationToCompletion}\hypertarget{RelationToCompletion}{}
\textbf{(relation to completion)}

A $\neg\{p\}$-local spectrum is also called \textbf{$\mathbb{Z}/p\mathbb{Z}$-acyclic}. According to the general theory of [[Bousfield localization of spectra]], they are ``dual'' to the ``$\mathbb{Z}/p\mathbb{Z}$-local spectra'', in the sense that $X$ is $\mathbb{Z}/p\mathbb{Z}$-local if every map $Y \to X$ out of a $\mathbb{Z}/p\mathbb{Z}$-acyclic $Y$ is [[null homotopy|null homotopic]].

$\mathbb{Z}/p\mathbb{Z}$-local spectra are also known as \emph{$p$-complete} spectra, and are the [[Bousfield localization of spectra]] at the [[Moore spectrum]] $S \mathbb{Z}/p\mathbb{Z}$ (e.g. \hyperlink{Bauer11}{Bauer 11, Example 1.7}).

This may be regarded as a consequence of the [[mod p Whitehead theorem]].

\end{remark}
See (\hyperlink{MayPonto}{May-Ponto, example 19.2.3}, \hyperlink{Lurie}{Lurie, example 8},\hyperlink{LurieProper}{Lurie ``Proper morphisms'', section 4}).

In summary:

[[!include localization and completion -- table]]

\begin{remark}
\label{}\hypertarget{}{}
In terms of [[arithmetic geometry]] this may be understood as follows:

\begin{enumerate}%
\item $\mathbb{F}_p = \mathbb{Z}/(p \mathbb{Z})$ is the [[ring of functions]] exactly on the point $(p)\in$ [[Spec(Z)]]


\item $\mathbb{Z}_p$ is the functions on the [[formal neighbourhood]] of $(p)$.


\item $\mathbb{Z}_{(p)}$ is the ring of functions defined on the open complement of the complement of $(p)$, hence on an [[open neighbourhood]].



\end{enumerate}
So the localization at the Moore spectrum of one of these rings localizes to the formal neighbourhood of their support. For $\mathbb{F}_p$ the support is the closed point, and so the localization there is infinitesimally bigger than that, as given by the $p$-adic completion. On the other hand the support of $\mathbb{Z}_{(p)}$ is open and hence contains its own formal neighbourhood, so localizing here just gives the plain localization.

\end{remark}
\hypertarget{basic_properties}{}\subsubsection*{{Basic properties}}\label{basic_properties}

\begin{itemize}%
\item [[fracture theorem]]

\end{itemize}
\hypertarget{in_chromatic_homotopy_theory}{}\subsubsection*{{In chromatic homotopy theory}}\label{in_chromatic_homotopy_theory}

Many of the basic constructions and theorems in [[chromatic homotopy theory]] apply to [[finite spectrum|finite]] $p$-local spectra, such as

\begin{itemize}%
\item [[telescopic localization]]


\item [[periodicity theorem]]


\item [[chromatic convergence theorem]]



\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[Brown-Peterson spectrum]]

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

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


\item [[finite homotopy type]], [[finite spectrum]]


\item [[nilpotent homotopy type]]


\item [[rational homotopy type]]



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

\begin{itemize}%
\item [[Dennis Sullivan]], \emph{Localization, Periodicity and Galois Symmetry} (The 1970 MIT notes) edited by [[Andrew Ranicki]], K-Monographs in Mathematics, Dordrecht: Springe (\href{http://www.maths.ed.ac.uk/~aar/surgery/gtop.pdf}{pdf})


\item [[Aldridge Bousfield]], [[Daniel Kan]], \emph{[[Homotopy limits, completions and localizations]]}, Lecture Notes in Mathematics, Vol 304, Springer 1972


\item [[Peter May]], [[Kate Ponto]], \emph{More concise algebraic topology: Localization, completion, and model categories} (\href{http://www.maths.ed.ac.uk/~aar/papers/mayponto.pdf}{pdf})


\item Carles Casacuberta and Georg Peschke, \emph{Localizing with respect to self maps of the circle} (\href{http://atlas.mat.ub.es/personals/casac/articles/cpes.pdf}{pdf}).


\item [[Jacob Lurie]], \emph{[[Chromatic Homotopy Theory]]}, Lecture series 2010, Lecture 20 \emph{Bousfield localization} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture20.pdf}{pdf})


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


\item [[Jacob Lurie]], section 4 of \emph{[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]}



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Localization_of_a_topological_space}{Localization of a topological space}}

\end{itemize}
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