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\begin{document}

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\section*{reflective 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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{ReflectiveLoc}{Reflective localization}\dotfill \pageref*{ReflectiveLoc} \linebreak
\noindent\hyperlink{ReflectionOntoLocalObjects}{Reflection onto local objects}\dotfill \pageref*{ReflectionOntoLocalObjects} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \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}

A [[localization of a category]]/[[localization of an (∞,1)-category|of an (∞,1)-category]] is called \textbf{reflective} if its [[localization functor]] has a [[fully faithful functor|fully faithful]] [[right adjoint]], hence if it is the reflector of a [[reflective subcategory]]/[[reflective sub-(∞,1)-category]]-inclusion.

In fact every [[reflective subcategory]] inclusion exhibits a reflective localization (Prop. \ref{ReflectiveSubcategoriesAreLocalizations} below).

For reflective localizations the localized category has a particularly useful description (Prop. \ref{ReflectiveLocalizationGivenByLocalObjects} below): It is equivalent to the [[full subcategory]] of \emph{[[local objects]]} (Def. \ref{LocalObjects} below).

Therefore, sometimes reflective localizations at a class $S$ or morphism are understood as the default concept of localization, in fact often reflection onto the [[full subcategory]] of $S$-[[local objects]] (Def. \ref{LocalizationAtACollectionOfMorphisms} below) is understood by default. Notably [[left Bousfield localizations]] are [[presentable (infinity,1)-category|presentations]] of reflective [[localizations of (∞,1)-categories]] in this sense.

These reflections onto $S$-local objects satisfy the universal property of an $S$-[[localization]] (only) for all \emph{[[left adjoint]]} functors that invert the class $S$ (Prop. \ref{ReflectionOntoLocalObjectsIsLocalizationWithRespectToLeftAdjoints} below).

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

\hypertarget{ReflectiveLoc}{}\subsubsection*{{Reflective localization}}\label{ReflectiveLoc}

\begin{defn}
\label{CategoryWithWeakEquivalences}\hypertarget{CategoryWithWeakEquivalences}{}
\textbf{([[category with weak equivalences]])}

A \emph{[[category with weak equivalences]]} is

\begin{enumerate}%
\item a [[category]] $\mathcal{C}$,


\item a [[subcategory]] $W \subset \mathcal{C}$



\end{enumerate}
such that the [[morphisms]] in $W$

\begin{enumerate}%
\item include all the [[isomorphisms]] of $\mathcal{C}$,


\item satisfy \emph{[[two-out-of-three]]}:

If for $g$, $f$ any two [[composition|composable]] [[morphisms]] in $\mathcal{C}$, two out of the set $\{g,\, f,\, g \circ f \}$ are in $W$, then so is the third.

\begin{displaymath}
\itexarray{
    & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}}
    \\
    && \underset{ g \circ f }{\longrightarrow}
  }
\end{displaymath}


\end{enumerate}
\end{defn}
\begin{defn}
\label{LocalizationOfACategory}\hypertarget{LocalizationOfACategory}{}
\textbf{([[localization of a category]])}

Let $W \subset \mathcal{C}$ be a [[category with weak equivalences]] (Def. \ref{CategoryWithWeakEquivalences}). Then the \emph{[[localization of a category|localization]]} of $\mathcal{C}$ at $W$ is, if it exists

\begin{enumerate}%
\item a [[category]] $\mathcal{C}[W^{-1}]$


\item a [[functor]] $\gamma \;\colon\; \mathcal{C} \longrightarrow \mathcal{C}[W^{-1}]$



\end{enumerate}
such that

\begin{enumerate}%
\item $\gamma$ sends all morphisms in $W \subset \mathcal{C}$ to [[isomorphisms]],


\item $\gamma$ is [[universal property|universal with this property]]: If $F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ is any functor with this property, then it factors through $\gamma$, up to [[natural isomorphism]]:

\begin{displaymath}
F \;\simeq\;
  D F \circ \gamma
  \phantom{AAAAAAA}
  \itexarray{
    \mathcal{C}
      && \overset{F}{\longrightarrow} &&
    \mathcal{D}
    \\
    & {}_{\mathllap{\gamma}}\searrow
      &{}^{\rho}\Downarrow_{\simeq}&
    \nearrow_{\mathrlap{D F}}
    \\
    && \mathcal{C}[W^{-1}]
  }
\end{displaymath}
and any two such factorizations $D F$ and $D^' F$ are related by a unique [[natural isomorphism]] $\kappa$ compatible with $\rho$ and $\rho^'$:



\end{enumerate}
\begin{equation}
\itexarray{
       \mathcal{C}
         && \overset{F}{\longrightarrow} &&
       \mathcal{D}
       \\
       &
       {}_{\mathllap{\gamma}}\searrow
         &{}^{\rho}\Downarrow_{\simeq}&
       \nearrow_{\mathrlap{D F}}
       && \searrow^{\mathrlap{id}}
       \\
       && \mathcal{C}[W^{-1}]
       && {}_{\simeq}\seArrow^{\kappa} && \mathcal{D}
       \\
       && &  {}_{\mathllap{id}}\searrow && \nearrow_{\mathrlap{D^' F}}
       \\
       && && \mathcal{C}[W^{-1}]
     }
     \phantom{AAAA}
     =
     \phantom{AAAA}
     \itexarray{
       \mathcal{C}
         && \overset{F}{\longrightarrow} &&
       \mathcal{D}
       \\
       & {}_{\mathllap{\gamma}}\searrow
         &{}^{\rho^'}\Downarrow_{\simeq}&
       \nearrow_{\mathrlap{D^' F}}
       \\
       && \mathcal{C}[W^{-1}]
     }
\label{CompatibleNaturalIsoForLocalization}\end{equation}
Such a localization is called a \emph{[[reflective localization]]} if the localization functor has a [[fully faithful functor|fully faithful]] [[right adjoint]], exhibiting it as the reflection functor of a [[reflective subcategory]]-inclusion

\begin{displaymath}
\mathcal{C}[W^{-1}]
  \underoverset
    {\underset{\phantom{AAAA}}{\hookrightarrow}}
    {\overset{ \phantom{AA} \gamma \phantom{AA} }{\longleftarrow}}
    {\bot}
  \mathcal{C}
\end{displaymath}
\end{defn}
\hypertarget{ReflectionOntoLocalObjects}{}\subsubsection*{{Reflection onto local objects}}\label{ReflectionOntoLocalObjects}

It turns out (Prop. \ref{ReflectiveLocalizationGivenByLocalObjects}) below, that reflective localizations at a collection $S$ of [[morphisms]] are, when they exist, [[reflective subcategory|reflections]] onto the [[full subcategory]] of \emph{$S$-[[local objects]]} (Def. \ref{LocalObjects} below). Often this reflection of $S$-[[local objects]] is what one is more interested in than the [[universal property]] of the $S$-[[localization]] according to (Def. \ref{CategoryWithWeakEquivalences}). This \emph{reflection onto local objects} (Def. \ref{LocalizationAtACollectionOfMorphisms} below) is what is often meant by default with ``localization'' (for instance in [[Bousfield localization]]).

\begin{defn}
\label{LocalObjects}\hypertarget{LocalObjects}{}
\textbf{([[local object]])}

Let $\mathcal{C}$ be a [[category]] and let $S \subset Mor_{\mathcal{C}}$ be a set of [[morphisms]]. Then an [[object]] $X \in \mathcal{C}$ is called an \emph{$S$-[[local object]]} if for all $A \overset{s}{\to} B \; \in S$ the [[hom-functor]] from $s$ into $X$ yields a [[bijection]]

\begin{displaymath}
Hom_{\mathcal{C}}(s,X)
  \;\colon\;
  Hom_{\mathcal{C}}(B,X)
   \overset{ \phantom{AA} \simeq \phantom{AA} }{\longrightarrow}
  Hom_{\mathcal{C}}(A,X)
  \,,
\end{displaymath}
hence if every morphism $A \overset{f}{\longrightarrow} X$ [[extension|extends]] uniquely along $s$ to $B$:

\begin{displaymath}
\itexarray{
    A &\overset{\phantom{A}f\phantom{A}}{\longrightarrow}& X
    \\
    {}^{\mathllap{s}}\big\downarrow & \nearrow_{\mathrlap{ \exists! }}
    \\
    B
  }
\end{displaymath}
We write

\begin{equation}
\mathcal{C}_S
  \overset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow}
  \mathcal{C}
\label{FullSubcategoryOfSLocalObjects}\end{equation}
for the [[full subcategory]] of $S$-local objects.

\end{defn}
\begin{defn}
\label{LocalizationAtACollectionOfMorphisms}\hypertarget{LocalizationAtACollectionOfMorphisms}{}
\textbf{([[reflective subcategory|reflection]] onto [[full subcategory]] of [[local objects]])}

Let $\mathcal{C}$ be a [[category]] and set $S \subset Mor_{\mathcal{C}}$ be a sub-[[class]] of its [[morphisms]]. Then the \emph{reflection onto local $S$-objects} (often called ``localization at the collection $S$'' is, if it exists, a [[left adjoint]] $L$ to the [[full subcategory]]-inclusion of the $S$-[[local objects]] \eqref{FullSubcategoryOfSLocalObjects}:

\begin{displaymath}
\mathcal{C}_S
    \underoverset
      {\underset{\iota}{\hookrightarrow}}
      {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}}
      {\bot}
  \mathcal{C}
  \,.
\end{displaymath}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{prop}
\label{ReflectiveSubcategoriesAreLocalizations}\hypertarget{ReflectiveSubcategoriesAreLocalizations}{}
\textbf{([[reflective subcategories]] are [[localizations]])}

Every [[reflective subcategory]]-inclusion

\begin{displaymath}
\mathcal{C}_{L}
  \underoverset
    {\underset{\phantom{AA}\iota \phantom{AA}}{\hookrightarrow}}
    {\overset{ \phantom{AA} L \phantom{AA} }{\longleftarrow}}
    {\bot}
  \mathcal{C}
\end{displaymath}
is [[generalized the|the]] [[reflective localization]] at the class $W \coloneqq L^{-1}(Isos)$ of morphisms that are sent to isomorphisms by the reflector $L$.

\end{prop}
\begin{proof}
Let $F \;\colon\; \mathcal{C} \to \mathcal{D}$ be a [[functor]] which inverts morphisms that are inverted by $L$.

First we need to show that it factors through $L$, up to natural isomorphism. But consider the following [[whiskering]] $F(\eta)$ of the [[adjunction unit]] $\eta$ with $F$:

\begin{displaymath}
\itexarray{
    \mathcal{C}
    && \overset{F}{\longrightarrow} &&
    \mathcal{D}
    \\
    & {}_{\mathllap{L}}\searrow &\Downarrow& \nearrow_{\mathrlap{D F}}
    \\
    && \mathcal{C}_L
  }
  \phantom{AA} \coloneqq \phantom{AA}
  \itexarray{
    \mathcal{C}
      && \overset{id}{\longrightarrow} &&
    \mathcal{C}
      & \overset{F}{\longrightarrow}&
    \mathcal{D}
    \\
    & {}_{\mathllap{L}}\searrow &\Downarrow^{\eta}& \nearrow_{\mathrlap{\iota}}
    \\
    && \mathcal{C}_L
  }
\end{displaymath}
By [[idempotent monad|idempotency]], the components of the [[adjunction unit]] $\eta$ are inverted by $L$, and hence by assumption they are also inverted by $F$, so that on the right the [[natural transformation]] $F(\eta)$ is indeed a [[natural isomorphism]].

It remains to show that this factorization is unique up to unique natural isomorphism. So consider any other factorization $D^' F$ via a natural isomorphism $\rho$. Pasting this now with the [[adjunction counit]]

\begin{displaymath}
\itexarray{
    &&
    \mathcal{C} && \overset{F}{\longrightarrow} &&  \mathcal{D}
    \\
    &
    {}^{\mathllap{\iota}}\nearrow
    &
    {}^{\epsilon}\Downarrow
    & {}_{\mathllap{L}}\searrow
    &\Downarrow^{\rho}& \nearrow_{\mathrlap{D^' F}}
    \\
    \mathcal{C}_L
    &&
      \underset{ id }{\longrightarrow}
    &&
    \mathcal{C}_L
  }
\end{displaymath}
exhibits a natural isomorphism $\epsilon \cdot \rho$ between $D F \simeq D^' F$. Moreover, this is compatible with $F(\eta)$ according to \eqref{CompatibleNaturalIsoForLocalization}, due to the [[triangle identity]]:

\begin{displaymath}
\itexarray{
    \mathcal{C}
    &&
     \overset{id}{\longrightarrow}
    &&
    \mathcal{C} && \overset{F}{\longrightarrow} &&  \mathcal{D}
    \\
    &
      {}_{\mathllap{id}}\searrow
    &
    {}^{\mathllap{\eta}}\Downarrow
    &
    {}^{\mathllap{\iota}}\nearrow
    &
    {}^{\epsilon}\Downarrow
    & {}_{\mathllap{L}}\searrow
    &\Downarrow^{\rho}& \nearrow_{\mathrlap{D^' F}}
    \\
    &&
    \mathcal{C}_L
    &&
      \underset{ id }{\longrightarrow}
    &&
    \mathcal{C}_L
  }
  \phantom{AAAA}
  =
  \phantom{AAAA}
  \itexarray{
    \mathcal{C} && \overset{F}{\longrightarrow} &&  \mathcal{D}
    \\
    & \searrow &\Downarrow^\rho& \swarrow
    \\
    && \mathcal{C}_L
  }
\end{displaymath}
Finally, since $L$ is [[essentially surjective functor]], by [[idempotent monad|idempotency]], it is clear that this is the unique such natural isomorphism.

\end{proof}
\begin{prop}
\label{ReflectiveLocalizationGivenByLocalObjects}\hypertarget{ReflectiveLocalizationGivenByLocalObjects}{}
\textbf{([[reflective localization]] [[reflective subcategory|reflects]] onto [[full subcategory]] of [[local objects]])}

Let $W \subset \mathcal{C}$ be a [[category with weak equivalences]] (Def. \ref{CategoryWithWeakEquivalences}). If its [[reflective localization]] (Def. \ref{LocalizationOfACategory}) exists

\begin{displaymath}
\mathcal{C}[W^{-1}]
    \underoverset
      {\underset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow}}
      {\overset{ \phantom{AA} L \phantom{AA} }{\longleftarrow}}
      {\bot}
  \mathcal{C}
\end{displaymath}
then $\mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C}$ is [[equivalence of categories|equivalently]] the inclusion of the [[full subcategory]] on the $W$-[[local objects]] (Def. \ref{LocalObjects}), and hence $L$ is equivalently reflection onto the $W$-local objects, according to Def. \ref{LocalizationAtACollectionOfMorphisms}.

\end{prop}
\begin{proof}
We need to show that

\begin{enumerate}%
\item every $X \in \mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C}$ is $W$-[[local object|local]],


\item every $Y \in \mathcal{C}$ is $W$-[[local object|local]] precisely if it is [[isomorphism|isomorphic]] to an object in $\mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C}$.



\end{enumerate}
The first statement follows directly with the [[adjoint functor|adjunction isomorphism]]:

\begin{displaymath}
Hom_{\mathcal{C}}(w, \iota(X))
    \simeq
  Hom_{\mathcal{C}[W^{-1}]}(L(w), X)
\end{displaymath}
and the fact that the [[hom-functor]] takes [[isomorphisms]] to [[bijections]].

For the second statement, consider the case that $Y$ is $W$-local. Observe that then $Y$ is also local with respect to the class

\begin{displaymath}
W_{sat} \;\coloneqq\; L^{-1}(Isos)
\end{displaymath}
of \emph{all} morphisms that are inverted by $L$ (the ``[[saturated class of morphisms]]''): For consider the [[hom-functor]] $\mathcal{C} \overset{Hom_{\mathcal{C}}(-,Y)}{\longrightarrow} Set^{op}$ to the [[opposite category|opposite]] of the [[category of sets]]. But assumption on $Y$ this takes elements in $W$ to isomorphisms. Hence, by the defining [[universal property]] of the [[localization]]-functor $L$, it factors through $L$, up to [[natural isomorphism]].

Since by [[idempotent monad|idempotency]] the [[adjunction unit]] $\eta_Y$ is in $W_{sat}$, this implies that we have a [[bijection]] of the form

\begin{displaymath}
Hom_{\mathcal{C}}( \eta_Y, Y )
  \;\colon\;
  Hom_{\mathcal{C}}( \iota L(Y), Y )
    \overset{\simeq}{\longrightarrow}
  Hom_{\mathcal{C}}(Y, Y)
  \,.
\end{displaymath}
In particular the [[identity morphism]] $id_Y$ has a [[preimage]] $\eta_Y^{-1}$ under this function, hence a [[left inverse]] to $\eta$:

\begin{displaymath}
\eta_Y^{-1} \circ \eta_Y \;=\; id_Y
  \,.
\end{displaymath}
But by [[2-out-of-3]] this implies that $\eta_Y^{-1} \in W_{sat}$. Since the first item above shows that $\iota L(Y)$ is $W_{sat}$-local, this allows to apply this same kind of argument again,

\begin{displaymath}
Hom_{\mathcal{C}}( \eta^{-1}_Y, \iota L(Y) )
    \;\colon\;
  Hom_{\mathcal{C}}( Y, \iota L(Y) )
    \overset{\simeq}{\longrightarrow}
  Hom_{\mathcal{C}}( \iota L(Y) , \iota L(Y))
  \,,
\end{displaymath}
to deduce that also $\eta_Y^{-1}$ has a [[left inverse]] $(\eta_Y^{-1})^{-1} \circ \eta_Y^{-1}$. But since a [[left inverse]] that itself has a [[left inverse]] is in fact an [[inverse morphisms]] (\href{retract#LeftInverseWithLeftInverseIsLeftInverse}{this Lemma}), this means that $\eta^{-1}_Y$ is an [[inverse morphism]] to $\eta_Y$, hence that $\eta_Y \;\colon\; Y \to \iota L (Y)$ is an [[isomorphism]] and hence that $Y$ is isomorphic to an object in $\mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C}$.

Conversely, if there is an [[isomorphism]] from $Y$ to a morphism in the image of $\iota$ hence, by the first item, to a $W$-local object, it follows immediatly that also $Y$ is $W$-local, since the [[hom-functor]] takes [[isomorphisms]] to [[bijections]] and since bijections satisfy [[2-out-of-3]].

\end{proof}
$\,$

\begin{itemize}%
\item For a left exact reflective localization, the class of morphisms that is inverted forms a left-[[multiplicative system]]. For the moment see at \emph{[[geometric embedding]]} for details on this.

\end{itemize}
\begin{prop}
\label{ReflectionOntoLocalObjectsIsLocalizationWithRespectToLeftAdjoints}\hypertarget{ReflectionOntoLocalObjectsIsLocalizationWithRespectToLeftAdjoints}{}
\textbf{([[reflective subcategory|reflection]] onto [[local objects]] in [[localization]] with respect to [[left adjoints]])}

Let $\mathcal{C}$ be a [[category]] and let $S \subset Mor_{\mathcal{C}}$ be a [[class]] of [[morphisms]] in $\mathcal{C}$. Then the [[reflective subcategory|reflection]] onto the $S$-[[local objects]] (Def. \ref{LocalizationAtACollectionOfMorphisms}) satisfies, if it exists, the [[universal property]] of a [[localization of categories]] (Def. \ref{LocalizationOfACategory}) with respect to \emph{[[left adjoint]]} functors inverting $S$.

\end{prop}
\begin{proof}
Write

\begin{displaymath}
\mathcal{C}_S 
    \underoverset
      {\underset{ \phantom{AA}\iota\phantom{AA} }{\hookrightarrow}}
      {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}}
      {\bot}
  \mathcal{C}
\end{displaymath}
for the [[reflective subcategory]]-inclusion of the $S$-[[local objects]].

Say that a [[morphism]] $f$ in $\mathcal{C}$ is an \emph{$S$-[[local morphism]]} if for every $S$-[[local object]] $A \in \mathcal{C}$ the [[hom-functor]] from $f$ to $A$ yields a [[bijection]] $Hom_{\mathcal{C}}(f,A)$. Notice that, by the [[Yoneda embedding]] for $\mathcal{C}_S$, the $S$-[[local morphisms]] are precisely the morphisms that are taken to isomorphisms by the reflector $L$.

Now let

\begin{displaymath}
(F \dashv G)
  \;\colon\;
  \mathcal{C} 
    \underoverset
      {\underset{G}{\longleftarrow}}
      {\overset{ \phantom{AA} F \phantom{AA} }{\longrightarrow}}
      {\bot}
   \mathcal{D}
\end{displaymath}
be a pair of [[adjoint functors]], such that the [[left adjoint]] $F$ inverts the morphisms in $S$. By the adjunction hom-isomorphism it follows that $G$ takes values in $S$-[[local objects]]. This in turn implies, now via the [[Yoneda embedding]] for $\mathcal{D}$, that $F$ inverts all $S$-[[local morphisms]], and hence all morphisms that are inverted by $L$.

Thus the essentially unique factorization of $F$ through $L$ now follows by Prop. \ref{ReflectiveSubcategoriesAreLocalizations}.

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

\begin{itemize}%
\item [[reflective subcategory]], [[coreflective subcategory]]


\item [[reflective factorization system]]


\item [[Bousfield localization]], [[Quillen reflection]]



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

The concept of reflective localization was originally highlighted in

\begin{itemize}%
\item [[Pierre Gabriel]], [[Michel Zisman]], \emph{[[Calculus of fractions and homotopy theory]]}, Springer 1967 (\href{https://www.math.rochester.edu/people/faculty/doug/otherpapers/GZ.pdf}{pdf})

\end{itemize}
A formalization in [[homotopy type theory]] of reflection onto local objects is discussed in

\begin{itemize}%
\item [[Egbert Rijke]], [[Michael Shulman]], [[Bas Spitters]], \emph{Modalities in homotopy type theory} (\href{https://arxiv.org/abs/1706.07526}{arXiv:1706.07526})

\end{itemize}
[[!redirects reflective localizations]]



\end{document}