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\begin{document}

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\section*{local object}

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

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

[[!include descent and locality - contents]]

\hypertarget{modalities_closure_and_reflection}{}\paragraph*{{Modalities, Closure and Reflection}}\label{modalities_closure_and_reflection}

[[!include modalities - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition_for_ordinary_categories}{Definition for ordinary categories}\dotfill \pageref*{definition_for_ordinary_categories} \linebreak
\noindent\hyperlink{local_objects}{Local objects}\dotfill \pageref*{local_objects} \linebreak
\noindent\hyperlink{local_morphisms}{Local morphisms}\dotfill \pageref*{local_morphisms} \linebreak
\noindent\hyperlink{definition_for_categories}{Definition for $(\infty,1)$-categories}\dotfill \pageref*{definition_for_categories} \linebreak
\noindent\hyperlink{local_objects_2}{Local objects}\dotfill \pageref*{local_objects_2} \linebreak
\noindent\hyperlink{local_morphisms_2}{Local morphisms}\dotfill \pageref*{local_morphisms_2} \linebreak
\noindent\hyperlink{definition_in_model_categories}{Definition in model categories}\dotfill \pageref*{definition_in_model_categories} \linebreak
\noindent\hyperlink{PropInModCat}{Properties}\dotfill \pageref*{PropInModCat} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{saturated_class_of_morphisms}{Saturated class of morphisms}\dotfill \pageref*{saturated_class_of_morphisms} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak


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

[[reflective localization|Reflective localizations]] of categories and higher categories in the sense of [[left adjoint]] functors $L : C \to C'$ to inclusions $C' \hookrightarrow C$ of full subcategories (as in particular for [[geometric embeddings]]) are characterized by the collection $S \subset Mor(C)$ of morphisms of $C$ which are sent by $L$ to isomorphisms, or more generally to equivalences, as well as by the collection of objects which are \emph{local} with respect to these morphisms, in that these morphisms behave as equivalences with respect to homming into objects.

\hypertarget{definition_for_ordinary_categories}{}\subsection*{{Definition for ordinary categories}}\label{definition_for_ordinary_categories}

\hypertarget{local_objects}{}\subsubsection*{{Local objects}}\label{local_objects}

Let $C$ be a [[category]] and $S$ a collection of [[morphisms]] in $C$. Then an [[object]] $c \in C$ is \textbf{$S$-local} if the [[hom-functor]]

\begin{displaymath}
C(-,c) : C^{op} \to Set
\end{displaymath}
sends morphisms in $S$ to [[isomorphisms]] in [[Set]], i.e. if for every $s : a \to b$ in $S$, the function

\begin{displaymath}
C(s,c) : C(b,c) \to C(a,c)
\end{displaymath}
is a [[bijection]].

\hypertarget{local_morphisms}{}\subsubsection*{{Local morphisms}}\label{local_morphisms}

Conversely, a \textbf{morphism} $f : x \to y$ is \textbf{$S$-local if for every $S$-local object $c$ the induced morphism}

\begin{displaymath}
C(f,c) : C(y,c) \to C(x,c)
\end{displaymath}
is an isomorphism.

\hypertarget{definition_for_categories}{}\subsection*{{Definition for $(\infty,1)$-categories}}\label{definition_for_categories}

\hypertarget{local_objects_2}{}\subsubsection*{{Local objects}}\label{local_objects_2}

\begin{defn}
\label{}\hypertarget{}{}
Let $C$ be an [[(∞,1)-category]] and $S$ a collection of morphisms in $C$. Then an [[object]] $c \in C$ is \textbf{$S$-local} if the [[hom-functor]]

\begin{displaymath}
C(-,c) : C^{op} \to \infty Top
\end{displaymath}
evaluated on $s \in S$ induces isomorphism in the [[homotopy category]] of [[Top]].

\end{defn}
This is \href{http://www.math.harvard.edu/~lurie/papers/highertopoi.pdf#page=383}{5.5.4.1} in [[Higher Topos Theory|HTT]]

\hypertarget{local_morphisms_2}{}\subsubsection*{{Local morphisms}}\label{local_morphisms_2}

Conversely, a \textbf{morphism} $f : x \to y$ is \textbf{$S$-local} if for every $S$-local object $c$ the induced morphism

\begin{displaymath}
C(f,c) : C(y,c) \to C(x,c)
\end{displaymath}
induces an isomorphism in the [[homotopy category]] of [[Top]].

\hypertarget{definition_in_model_categories}{}\subsection*{{Definition in model categories}}\label{definition_in_model_categories}

Let $C$ be a [[model category]] (usefully but not necessarily a [[simplicial model category]]). And let $S \subset Mor(C)$ be a collection of [[morphism]]s in $C$.

Write $\mathbf{R}Hom_C(-,-) : C^{op}\times C \to SSet$ for the [[(infinity,1)-categorical hom-space|derived hom space functor]].

For instance if $C$ is a [[simplicial model category]] then this may be realized in terms of a cofibrant replacement functor $Q : C \to C$ and a fibrant replacement functor $P$ as

\begin{displaymath}
\mathbf{R}Hom_C(X,Y) = C(Q X, P Y)
  \,.
\end{displaymath}
\begin{defn}
\label{}\hypertarget{}{}
\textbf{(local object, local weak equivalence)}

An object $c \in C$ is a \textbf{$S$-local object} if for all $s : a \to b$ in $S$ the induced morphism

\begin{displaymath}
\mathbf{R}Hom_C(s,c) : \mathbf{R}Hom_C(b,c) \to \mathbf{R}Hom_C(a,c)
\end{displaymath}
is a weak equivalence (in the standard [[model structure on simplicial sets]]);

A morphism $f : x \to y$ in $C$ is an \textbf{$S$-local morphism} or \textbf{$S$-equivalence} if for every $S$-local object $c$ the induced morphism

\begin{displaymath}
\mathbf{R}Hom_C(f,c) : SSet \to SSet
\end{displaymath}
is a weak equivalence.

An \textbf{$S$-localization of an object} $c$ is an $S$-local object $\hat c$ and an $S$-local equivalence $c \to \hat c$.

An \textbf{$S$-localization of a morphism} $f : c \to d$ is a pair of $S$-localizations $c \to \hat c$ and $d \to \hat d$ of objects, and a commuting square

\begin{displaymath}
\itexarray{
    c &\stackrel{f}{\to}& d
    \\
    \downarrow && \downarrow
    \\
    \hat c &\to & \hat d
  }
  \,.
\end{displaymath}
\end{defn}
\hypertarget{PropInModCat}{}\subsubsection*{{Properties}}\label{PropInModCat}

In [[proper model category|left proper model categories]] there is an equivalent stronger characterization of $S$-locality of cofibrations $i : A \hookrightarrow B$.

\begin{prop}
\label{}\hypertarget{}{}
\textbf{(characterization of $S$-local cofibrations)}

Let $C$ be a [[proper model category|left proper]] [[simplicial model category]] and $S \subset Mor(C)$, a collection of morphisms.

Then a cofibration $i : A \hookrightarrow B$ is an $S$-local weak equivalence precisely if for all fibrant $S$-local objects $X$ the morphism

\begin{displaymath}
C(B,X) \to C(A,X)
\end{displaymath}
is an acyclic fibration in the standard [[model structure on simplicial sets]].

\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
Notice that this is stronger than the statement that $\mathbf{R}Hom(B,X) \to \mathbf{R}Hom(A,X)$ is a weak equivalence not only in that it asserts in addition a fibration, but also in that it deduces this without first passing to a cofibrant replacement of $A$ and $B$.

\end{remark}
\begin{proof}
This is [[Higher Topos Theory|HTT, lemma A.3.7.1]].

The proof makes use of the following general construction: for $f : A \to B$ any morphism let $\emptyset \hookrightarrow A' \stackrel{\simeq}{\to} A$ be a cofibrant replacement, factor $A' \to B$ as $A' \stackrel{i'}{\hookrightarrow} B' \stackrel{\simeq}{\to} B$ and consider the [[pushout]] diagram

\begin{displaymath}
\itexarray{
    A' &\stackrel{i'}{\hookrightarrow}& B'
    \\
    \downarrow^{\mathrlap{f \in W}}
    && \downarrow_{\mathrlap{g\in W}}
    &
    \searrow^{\mathrlap{f' \in W}}
    \\
    A &\stackrel{}{\hookrightarrow}& A \coprod_{A'} B 
    &\stackrel{j \in W}{\to}&
    B
  }
  \,.
\end{displaymath}
By [[proper model category|left properness]] the pushout $g$ of the weak equivalence $f$ along the cofibration $i'$ is again a weak equivalence and by [[category with weak equivalences|2-out-of-3]] the morphism $j$ is a weak equivalence.

Now assume that $i$ is an $S$-local equivalence. We need to show that $i^* : C(B,X) \to C(A,X)$ is an acyclic Kan fibration for all fibrant $S$-local $X$. By the very definition of [[enriched model category]] it follows from $i$ being a cofibration and $X$ being fibrant that this is a Kan fibration. So it remains to show that it is a [[weak homotopy equivalence]] of [[simplicial set]]s. We know that the corresponding induced morphism

\begin{displaymath}
({i'}^* : C(B',X) \to C(A',X))
  \simeq
  (\mathbf{R}Hom(B,X) \to \mathbf{R}Hom(A,X))
\end{displaymath}
on the cofibrant replacement is a weak equivalence, by the assumption that $X$ is $S$-local, and also, as before, a fibration, since $i'$ is still a cofibration.

By homming the entire diagram above into $X$, and using that the [[hom-functor]] $C(-,X)$ sends [[colimit]]s to [[limit]]s, we find the [[pullback]] diagram

\begin{displaymath}
\itexarray{
    C(A \coprod_{A'} B', X)
    &\to&
    C(B',X)
    \\
    {}^{q}\downarrow^{\mathrlap{\in (W\cap fib)_{SSet}}}
    &&
    {}^{{i'}^*}\downarrow^{\mathrlap{\in (W\cap fib)_{SSet}}}
    \\
    C(A,X) 
    &\to&
    C(A',X)
  }
\end{displaymath}
in [[SSet]], which shows that $q$ is an acyclic fibration, being the pullback of an acyclic fibration.

To show that $i^*: C(B,X) \to C(A,X)$ is a weak equivalence it suffices to show that all its fibers $(i^*)^{-1})(t)$ over elements $t : A \to X$ are [[contractible]] [[Kan complex]]es. These fibers map to the corresponding fibers $q^{-1}(t)$ by precomposition with $j$. By the fact that $j$, regarded as a morphism

\begin{displaymath}
\itexarray{
   && A
   \\
   & {}\swarrow 
    && \searrow
   \\
   A \coprod_{A'} B' &&\stackrel{j}{\to}&& B
  }
\end{displaymath}
in the [[model structure on an over category|model structure on the undercategory]] $A/C$ is a weak equivalence between cofibrant objects (because $A \hookrightarrow B$ is a cofibration by assumption and $A \to A \coprod_{A'} B'$ as being the pushout of the cofibration $i'$) we have that precomposition $C(j,X)$ with $j$ is the image under the [[SSet]]-[[enriched functor|enriched]] [[hom-functor]] of a weak equivalence between cofibrant objects mapping into a fibrant object

\begin{displaymath}
\itexarray{
    && A
    \\
    & \swarrow & \downarrow & \searrow^{t}
    \\
    A \coprod_{A'} B' 
    &\stackrel{j}{\to}&
    B
    &\to&
    X
  }
\end{displaymath}
and hence, by the general properties of \href{http://ncatlab.org/nlab/show/%28infinity,1%29-categorical+hom-space#enriched_homs_between_cofibrantfibrant_objects_6}{enriched homs between cofibrant/fibrant objects} a weak equivalence. $j^* : (i^*)^{-1}(t) \stackrel{\simeq}{\to} q^{-1}(t)$, so that indeed $(i^*)^{-1}(t)$ is contractible.

This proves the first part of the statement. For the converse statement, assume now that\ldots{}

\end{proof}
\hypertarget{references}{}\subsubsection*{{References}}\label{references}

A classical textbook reference is section 3.2 of

\begin{itemize}%
\item Hirschhorn, \emph{Model categories and their localization}

\end{itemize}
A useful reference with direct ties to the [[(∞,1)-category]] story in the background is section A.3.7 of

\begin{itemize}%
\item [[Jacob Lurie]], [[Higher Topos Theory]]

\end{itemize}
\hypertarget{saturated_class_of_morphisms}{}\subsection*{{Saturated class of morphisms}}\label{saturated_class_of_morphisms}

Every morphism in $S$ is $S$-local.

The collection $S$ of morphisms is called \textbf{saturated} if the collection of $S$-local morphisms coincides with $S$.

\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\begin{itemize}%
\item a [[reflective subcategory]] as well as a [[reflective (∞,1)-subcategory]] can be realized as the full ($(\infty,1)$-)subcategory on $S$-local objects, where $S$ is the collection of morphisms sent by the corresponding [[localization of an (∞,1)-category]] to equivalences. For details on this see the discussion at [[geometric embedding]].

\end{itemize}
[[!redirects local objects]] [[!redirects local morphism]] [[!redirects local morphisms]]



\end{document}