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\section*{localization}

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

\hypertarget{locality_and_descent}{}\paragraph*{{Locality and descent}}\label{locality_and_descent}

[[!include descent and locality - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{motivation}{Motivation}\dotfill \pageref*{motivation} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{reflective_localization}{Reflective localization}\dotfill \pageref*{reflective_localization} \linebreak
\noindent\hyperlink{localizations_of_enriched_categories}{Localizations of enriched categories}\dotfill \pageref*{localizations_of_enriched_categories} \linebreak
\noindent\hyperlink{construction}{Construction}\dotfill \pageref*{construction} \linebreak
\noindent\hyperlink{general_construction}{General construction}\dotfill \pageref*{general_construction} \linebreak
\noindent\hyperlink{construction_when_there_is_a_calculus_of_fractions}{Construction when there is a calculus of fractions}\dotfill \pageref*{construction_when_there_is_a_calculus_of_fractions} \linebreak
\noindent\hyperlink{in_abelian_categories}{In abelian categories}\dotfill \pageref*{in_abelian_categories} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{in_higher_category_theory}{In higher category theory}\dotfill \pageref*{in_higher_category_theory} \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 general, [[localization]] is a process of adding formal inverses to an algebraic structure. The \emph{localization} of a [[category]] $C$ at a collection $W$ of its [[morphism]]s is -- if it exists -- the result of universally making all morphisms in $W$ into [[isomorphism]]s.

\hypertarget{motivation}{}\subsubsection*{{Motivation}}\label{motivation}

A classic example is the [[localization of a commutative ring]]: we can `localize the ring $\mathbb{Z}$ away from the prime $2$' and obtain the ring $\mathbb{Z}[\frac{1}{2}]$, or localize it away from all primes and obtain its field of fractions: the field $\mathbb{Q}$ of [[rational number|rational numbers]].

The terminology is odd for historical and geometric reasons: localizing \emph{at} a prime means inverting things \emph{not} divisible by that prime, while inverting the prime itself is called localizing \emph{away} from that prime. The reason for this, as well as for the term `localization', becomes more apparent when we consider examples of a more vividly geometric flavor.

For example, the ring $\mathbb{R}[x]$ consists of [[polynomial]] functions on the real line. If we pick a point $a \in \mathbb{R}$ and localize $\mathbb{R}[x]$ by putting in an inverse to the element $(x-a)$, the resulting ring consists of [[rational function]]s defined \emph{everywhere} on the real line \emph{except} possibly at the point $a$. This is called \textbf{localization \emph{away from} $a$}, or localization away from the [[ideal]] $I$ generated by $(x-a)$.

If on the other hand we put in an inverse to every element of $\mathbb{R}[x]$ that is \emph{not} in the ideal $I$, we obtain the ring of rational functions defined \emph{somewhere} on the real line \emph{at least} at the point $a$: namely, those without a factor of $(x-a)$ in the denominator. This is called \textbf{localizing \emph{at} $a$}, or localizing at the ideal $I$.

Notice that what is literally `localized' when localizing the ring is not the ring itself, but its [[spectrum of a commutative ring|spectrum]]: the spectrum becomes smaller. The spectrum of $\mathbb{R}[x]$ is the whole real line. When we localize away from $a$, the resulting ring has spectrum $\mathbb{R} - \{a\}$. When we localize at $a$, the resulting ring has spectrum $\{a\}$.

The case of localizing $\mathbb{Z}$ can also be interpreted geometrically in a similar way, using [[scheme]] theory and [[arithmetic geometry]].

A ring is a very special case of a [[category]], namely a one-object [[Ab-enriched category]]. This article mainly treats the more general case of localizing an arbitrary category. The localization of a category $C$ at a class of [[morphisms]] $W$ is the universal solution to making the morphisms in $W$ into [[isomorphisms]]; it is variously written $C[W^{-1}]$, $W^{-1}C$ or $L_W C$. In some contexts, it also could be called the [[homotopy category]] of $C$ with respect to $W$.

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

Let $C$ be a [[category]] and $W \subset Mor(C)$ a collection of [[morphisms]].

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

A \textbf{localization} of $C$ by $W$ (or ``at $W$'') is

\begin{itemize}%
\item a (generally [[large category|large]], see below) [[category]] $C[W^{-1}]$;


\item and a [[functor]] $Q : C \to C[W^{-1}]$;


\item such that

\begin{itemize}%
\item for all $w \in W$, $Q(w)$ is an [[isomorphism]];


\item for any category $A$ and any [[functor]] $F : C \to A$ such that $F(w)$ is an [[isomorphism]] for all $w \in W$, there exists a functor $F_W : C[W^{-1}] \to A$ and a [[natural transformation|natural isomorphism]] $F \simeq F_W \circ Q$;


\item the map between [[functor category|functor categories]]

\begin{displaymath}
(-)\circ Q : Funct(C[W^{-1}], A) \to Funct(C,A)
\end{displaymath}
is [[fully faithful functor|full and faithful]] for every category $A$.



\end{itemize}


\end{itemize}
Note:

\begin{itemize}%
\item if $C[W^{-1}]$ exists, it is unique up to [[equivalence]].


\item In 2-categorical language, $C[W^{-1}]$ is the [[coinverter]] of the canonical natural transformation $s\to t$, where $s,t:W\to C$ are the ``source'' and ``target'' functors and $W$ is considered as a full subcategory of the [[arrow category]] $C ^{\mathbf{2}}$.



\end{itemize}
\begin{remark}
\label{}\hypertarget{}{}
Contrary to what is sometimes asserted, the localization $C[W^{-1}]$ may generally be constructed when $C$ is large, even only in ZF where ``large'' means definable by class formulas. See Remark \ref{ZF} below. However, the localization might not be locally small, even if $C$ is. The tools of [[homotopy theory]], and in particular [[model category|model categories]], can be used to address this issue (see also at \emph{[[homotopy category of a model category]]}).

\end{remark}
\hypertarget{reflective_localization}{}\subsubsection*{{Reflective localization}}\label{reflective_localization}

A special class of localizations are \emph{[[reflective localizations]]}, those where the functor $C \to L_W C$ has a [[full and faithful functor|full and faithful]] [[right adjoint]] $L_W C \hookrightarrow C$.

In such a case

\begin{displaymath}
L_W C 
    \stackrel{\overset{Q}{\leftarrow}}{\hookrightarrow}
  C
\end{displaymath}
this adjoint exhibits $L_W C$ as a [[reflective subcategory]] of $C$.

One shows that $L_W C$ is -- up to [[equivalence of categories]] -- the full subcategory on the $W$-[[local object]]s, and this property precisely characterizes such reflective localizations.

More on this is at \emph{[[reflective localization]]}, \emph{[[reflective subcategory]]}, \emph{[[reflective sub-(∞,1)-category]]}, and [[reflective factorization system]].

\hypertarget{localizations_of_enriched_categories}{}\subsubsection*{{Localizations of enriched categories}}\label{localizations_of_enriched_categories}

Given a symmetric closed monoidal category $V$, a $V$-enriched category $A$ with underlying ordinary category $A_0$ and a subcategory $\Sigma$ of $A_0$ containing the identities of $A_0$, H. Wolff defines the corresponding theory of localizations. See [[localization of an enriched category]].

\hypertarget{construction}{}\subsection*{{Construction}}\label{construction}

There is a general construction of $C[W^{-1}]$, if it exists, which is however hard to use. When the system $W$ has special properties, most notably when $W$ admits a [[calculus of fractions]] or a [[factorization system]], then there are more direct formulas for the [[hom-sets]] of $C[W^{-1}]$.

\hypertarget{general_construction}{}\subsubsection*{{General construction}}\label{general_construction}

If $C$ is a category and $W$ is a set of arrows, we construct the localization of $C$. Let $W^{op}$ be the set in $C^{op}$ corresponding to $W$ (it isn't necessarily a category).

Let $G$ be the following [[quiver|directed graph]]:

\begin{itemize}%
\item the vertices of $G$ are the objects of $C$,
\item the arrows of $G$ between two vertices $x,y$ are given by the disjoint union $C(x,y)\coprod W^{op}(x,y)$.

\end{itemize}
The arrows in $W^{op}(x,y)$ are written as $\overline{f}$ for $f\in W(y,x)$.

Let $\mathcal{P}G$ be the [[free functor|free]] category on $G$. The identity arrows are given by the empty path beginning and ending at a given object. We introduce a relation on the arrows of $\mathcal{P}G$ and quotient by the equivalence relation $\sim$ generated by it to get $C[W^{-1}]$.

The equivalence relation $\sim$ is generated by

\begin{itemize}%
\item for all objects $x$ of $C$,\begin{displaymath}
(x;id_x;x) \sim (x;\emptyset;x)
\end{displaymath}

\item for all $f:x\to y$ and $g:y\to z$ in $C$,\begin{displaymath}
(x;f,g;z)\sim (x;g\circ f;z)
\end{displaymath}

\item for all $f:x\to y$ in $W$,\begin{displaymath}
(x;f,\overline{f};x)\sim (x;id_x;x)
\end{displaymath}
and

\begin{displaymath}
(y;\overline{f},f;y)\sim (y;id_y;y)
\end{displaymath}


\end{itemize}
\begin{remark}
\label{ZF}\hypertarget{ZF}{}
With a little care, this general construction can be enacted even for \emph{large} categories $C$, where ``large'' means given by a class formula in [[ZF]]. First, to define the free category $\mathcal{P} G$ on a large graph $G$, notice that an arrow of $\mathcal{P} G$ is a certain [[partial function]] or functional relation from $\mathbb{N}$ to $Arr(G)$ satisfying some conditions given by ZF formulas; each such relation is a ``set'' by the [[replacement axiom]], and the collection of such relations forms a definable class of sets. Second, in passing to the quotient, we remark that while the $\sim$-equivalence classes of morphisms in $\mathcal{P} G$ might be large (hence themselves could not be elements of a class), we can always use [[Scott's trick]] and consider instead those representatives in each equivalence class that have minimal ZF [[rank]] (using crucially the [[axiom of foundation]]); this collection of representatives is a \emph{set}. Thus the collection of morphisms in the quotient may be identified with a definable class of sets.

The rub is that the localization might not be locally small, even if $C$ is.

\end{remark}
\hypertarget{construction_when_there_is_a_calculus_of_fractions}{}\subsubsection*{{Construction when there is a calculus of fractions}}\label{construction_when_there_is_a_calculus_of_fractions}

If the class $W$ admits a [[calculus of fractions]], then there is a simpler description of $C[W^{-1}]$ in terms of [[span|spans]] instead of zig-zags. The idea is that any morphism $f: x \to z$ in $C[W^{-1}]$ is built from a morphism $f_2 : y \to z$ in $C$ and a morphism $f_1 : y \to x$ in $W$:

\begin{displaymath}
x \stackrel{f_1}{\longleftarrow} y \stackrel{f_2}{\longrightarrow} z
\end{displaymath}
For more on this, see the entry [[calculus of fractions]].

Dorette Pronk has extended this idea to construct a [[bicategory of fractions|bicategories of fractions]] where a class of 1-arrows is sent to [[equivalence|equivalences]].

\hypertarget{in_abelian_categories}{}\subsubsection*{{In abelian categories}}\label{in_abelian_categories}

Localization is especially well developed in abelian setup where several competing formalisms and input data are used. See [[localization of an abelian category]].

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

\begin{itemize}%
\item The localization of the [[product category]] of two [[categories with weak equivalences]] is (if it exists) the product of their localizations

\begin{displaymath}
(\mathcal{C} \times \mathcal{D})[ (W_{\mathcal{C}} \times W_{\mathcal{D}})^{-1} ]
  \;\;
    \simeq
  \;\;
  ( \mathcal{C}[W_{\mathcal{C}}^{-1}] )
   \times
  ( \mathcal{D}[W_{\mathcal{D}}^{-1}] )
  \,.
\end{displaymath}
This is because localization is a [[reflector]] into the [[exponential ideal]] of minimal categories with weak equivalences.



\end{itemize}
\hypertarget{in_higher_category_theory}{}\subsection*{{In higher category theory}}\label{in_higher_category_theory}

The notion of localization of a category has analogs in [[higher category theory]].

For [[2-categories]]:

\begin{itemize}%
\item [[2-localization]]

\end{itemize}
For [[(∞,1)-categories]] and the special case of reflective embeddings this is discussed in

\begin{itemize}%
\item [[localization of an (∞,1)-category]].

\end{itemize}
Every [[locally presentable (∞,1)-category]] is presented by a [[combinatorial model category|combinatorial]] [[model category]]. Accordingly, there is a model for the localization of $(\infty,1)$-categories in terms of these models. This is called

\begin{itemize}%
\item left [[Bousfield localization of model categories]]

\end{itemize}
See also [[localization of a simplicial model category]].

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

\begin{itemize}%
\item [[localizing subcategory]], [[localizer]]


\item [[localization at an object]]


\item [[monoidal localization]]


\item [[simplicial localization]]


\item [[localization of a simplicial model category]]


\item [[localization of a ring]], [[localization of a commutative ring]]

\begin{itemize}%
\item [[localization of a module]]

\end{itemize}

\item [[localization of model categories]]

\begin{itemize}%
\item [[localization of simplicial model categories]]


\item [[Bousfield localization of model categories]]

\begin{itemize}%
\item [[Bousfield localization of spectra]]


\item [[p-localization]]



\end{itemize}


\end{itemize}

\item [[colocalization]]



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

The classical reference to localization for categories is the book by Gabriel and Zisman:

\begin{itemize}%
\item P. Gabriel, M. Zisman, \emph{[[Calculus of fractions and homotopy theory]]}, Springer, New York, 1967. \emph{Ergebnisse der Mathematik und ihrer Grenzgebiete}, Band 35.

\end{itemize}
A more recent account of localization with a [[calculus of fractions]] is section 7 of

\begin{itemize}%
\item [[M. Kashiwara]], [[P. Schapira]], \emph{[[Categories and Sheaves]]}, Springer 2000.

\end{itemize}
An excellent account emphasizing the interplay of the different notions (reflective subcategory, calculus of fractions, closure operator) can be found in ch. V of

\begin{itemize}%
\item [[Francis Borceux|F. Borceux]], \emph{Handbook of Categorical Algebra vol. 1} , Cambridge UP 1994.\footnote{See also vols.2,3 for examples of the theory in action in abelian categories, sheaf theory etc.} 

\end{itemize}
The pioneering work on abelian categories, with a large part on the localization in abelian categories is

\begin{itemize}%
\item [[Pierre Gabriel]], [[Des catégories abéliennes]], Bulletin de la Soci\'e{}t\'e{} Math\'e{}matique de France, 90 (1962), p. 323-448, \href{http://www.numdam.org/item?id=BSMF_1962__90__323_0}{numdam}

\end{itemize}
A \textbf{terminological discussion} prompted by question in which sense ``localization'' is a descriptive term or not is archived ion $n$Forum \href{https://nforum.ncatlab.org/discussion/481/localization/?Focus=23461#Comment_23461}{here}.

A formal implementation of Gabriel-Zisman localization in [[ZFC]], which in turn is implemented in the [[proof assistant]] [[Coq]] is in

\begin{itemize}%
\item [[Carlos Simpson]], \emph{Explaining Gabriel-Zisman localization to the computer} (\href{http://arxiv.org/abs/math/0506471}{arXiv:math/0506471}, \href{http://math.unice.fr/~carlos/themes/verif.html}{web})

\end{itemize}
A [[HoTT]]-[[Coq]]-formalization of left-exact [[reflective sub-(∞,1)-categories]] ([[localization of an (∞,1)-category]]) in [[homotopy type theory]] is in

\begin{itemize}%
\item [[Mike Shulman]], \emph{\href{https://github.com/mikeshulman/HoTT/blob/master/Coq/Subcategories/LexReflectiveSubcategory.v}{HoTT/Coq/Subcategories/LexReflectiveSubcategory.v}}

\end{itemize}
An original account of [[localization of commutative rings]] and of [[p-localization|p-local]] [[homotopy theory]] is

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

\end{itemize}
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