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\begin{document}

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\section*{topological localization}

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

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory}

[[!include (infinity,1)-topos - 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{for_presheaf_categories}{For $(\infty,1)$-presheaf $(\infty,1)$-categories}\dotfill \pageref*{for_presheaf_categories} \linebreak
\noindent\hyperlink{model_category_presentation}{Model category presentation}\dotfill \pageref*{model_category_presentation} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \textbf{topological localization} is a left exact [[localization of an (∞,1)-category]] -- in the sense of passing to a [[reflective sub-(∞,1)-category]] -- at a collection of morphisms that are [[monomorphism in an (∞,1)-category|monomorphisms]].

A topological localization of an [[(∞,1)-category of (∞,1)-presheaves]] $PSh_{(\infty,1)}(C)$ is precisely a localization at [[Cech cover]]s for a given [[Grothendieck topology]] on $C$, yielding the corresponding [[(∞,1)-topos]] [[(∞,1)-category of (∞,1)-sheaves|of (∞,1)-sheaves]].

\begin{displaymath}
Sh_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C)
\end{displaymath}
and in fact equivalence classes of such topological localizations are in bijection with [[Grothendieck topology|Grothendieck topologies]] on $C$.

Notice that in general a topological localization is not a [[hypercomplete (∞,1)-topos]]. That in general requires localization further at [[hypercover]]s.

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

Recall that a [[reflective sub-(∞,1)-category]] $D \stackrel{\stackrel{L}{\leftarrow}}{\hookrightarrow} C$ is obtained by localizing at a collection $S$ of morphisms of $C$.

The class $\bar S$ of all morphisms of $C$ that the left adjoint $L : C \to D$ sends to equivalences is the \emph{strongly saturated class} of morphisms generated by $S$. By the , the functor $L$ is exact if and only if $\bar S$ is stable under the formation of pullbacks.

We now define such localizations where the collection $S$ consists of \emph{[[monomorphism in an (∞,1)-category|monomorphisms]]} .

\begin{defn}
\label{}\hypertarget{}{}
Call a morphism $f : X \to Y$ in an [[(∞,1)-category]] $C$ a \textbf{[[monomorphism in an (∞,1)-category|monomorphism]]} if it is a [[n-truncated object of an (infinity,1)-category|(-1)-truncated object]] in the [[overcategory]] $X_{/Y}$.

Equivalently: if for every object $A \in C$ the induced morphism in the [[homotopy category]] of [[∞-groupoid]]s

\begin{displaymath}
C(A,f) : C(A,X) \to C(A,Y)
\end{displaymath}
exhibits $C(A,X)$ as a [[direct sum|direct summand]] of $C(A,Y)$.

Equivalence classes of monomorphisms into an object $X$ form a [[poset]] $Sub(X)$ of [[subobject]]s of $X$.

\end{defn}
This is [[Higher Topos Theory|HTT, p. 460]]

\begin{example}
\label{}\hypertarget{}{}
The standard example to keep in mind is that of a [[Cech nerve]]. In fact, as the propositions below will imply, this is for the purposes of localizations of an [[(∞,1)-category of (∞,1)-presheaves]] the \emph{only} kind of example.

Let [[Diff]] be the category of smooth manifolds and $PSh_{(\infty,1)}(Diff)$ the [[(∞,1)-category of (∞,1)-presheaves]] on $Diff$, which may be modeled by the global [[model structure on simplicial presheaves]] on $Diff$.

For $X \in Diff$ a [[manifold]], let $\{U_i \hookrightarrow X\}$ be an [[open cover]]. Let $C(\{U_i\})$ be the [[Cech nerve]] of this cover, the [[simplicial object]] of presheaves

\begin{displaymath}
C(\{U_i\}) = 
  \left(
     \cdots
     \coprod_{i j} U_i \cap U_j  \stackrel{\to}{\to}\coprod_{i} U_i   
  \right)
  \,.
\end{displaymath}
which we may regard as a [[simplicial presheaf]] and hence as an object of $PSh_{(\infty,1)}(Diff)$.

Then for $V$ any other [[manifold]], we have that

\begin{displaymath}
PSh_{(\infty,1)}(V, C(\{U_i\}))
\end{displaymath}
is the [[∞-groupoid]] whose

\begin{itemize}%
\item [[object]]s are maps $V \to X$ that factor through one of the $U_i$;


\item there is a unique [[morphism]] between two such maps precisely if they factor through a double intersection $U_{i} \cap U_j$;


\item and so on.



\end{itemize}
In the [[homotopy category of an (∞,1)-category|homtopy category]] of [[∞-groupoid]]s, this is equivalent to the [[0-groupoid]]/[[set]] of those maps $V \to X$ that factor through one of the $U_i$. Notice that this constitutes the [[sieve]] generated by the [[covering]] family $\{U_i \to X\}$. This is a subset of the [[0-groupoid]]/[[set]] $PSh_{(\infty)}(V,X) = Hom_{Diff}(V,X)$, hence a [[direct sum]]mand.

\end{example}
\begin{defn}
\label{}\hypertarget{}{}
\textbf{topological localization}

Let $C$ be a [[presentable (∞,1)-category]].

A strongly saturated class $\bar S \subset Mor(C)$ of morphisms is called \textbf{topological} if

\begin{itemize}%
\item there is a subclass $S \subset \bar S$ of \emph{[[monomorphism in an (infinity,1)-category|monomorphisms]]} that generates $\bar S$;


\item under [[pullback]] in $C$ elements in $\bar S$ pull back to elements in $\bar S$.



\end{itemize}
A [[reflective sub-(∞,1)-category]]

\begin{displaymath}
D \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} C
\end{displaymath}
is called a \textbf{topological localization} if the class of morphisms $\bar S := L^{-1}(equiv)$ that $L$ sends to equivalences is topological.

\end{defn}
This is [[Higher Topos Theory|HTT, def. 6.2.1.4]]

\begin{defn}
\label{}\hypertarget{}{}
\textbf{$(\infty,1)$-sheaves}

Let $C$ be an [[(∞,1)-site]].

Let $S$ be the collection of all [[monomorphism in an (∞,1)-category|monomorphisms]] $U \to c$ to objects $c \in Y$ (under [[(infinity,1)-Yoneda lemma|Yoneda embedding]]) that correspond to [[covering]] [[sieve]]s in $C$. Say an object $c \in PSh_{(\infty,1)}(C)$ in the [[(∞,1)-category of (∞,1)-presheaves]] on $C$ is an \textbf{[[(∞,1)-sheaf]]} if it is an $S$-[[local object]] (i.e. if it satisfies [[descent]] along all morphisms $U \to c$ coming from covering sieves).

Write

\begin{displaymath}
Sh_{(\infty,1)}(C)  \hookrightarrow PSh_{(\infty,1)}(X)
\end{displaymath}
for the [[reflective sub-(∞,1)-category]] on these $(\infty,1)$-sheaves.

\end{defn}
This is [[Higher Topos Theory|HTT, def. 6.2.2.6]]

\textbf{Warning:} A topological localization is, by definition, a \emph{left exact} localization at a set of monomorphisms; left exactness is part of the definition. A general localization at a set of monomorphisms need not be left exact. For instance, the localization at one of the inclusions $1\to 1+1$ is the $(-1)$-truncation, which is not left exact.

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

Let throughout $C$ be a [[locally presentable (∞,1)-category]].

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

\begin{cor}
\label{}\hypertarget{}{}
\textbf{(topological localizations are exact)}

Every topological localization is an \emph{exact localization} in that the reflector $L : C \to D$ preserves [[finite limit]]s.

\end{cor}
\begin{proof}
At  it is shown that a reflective localization is exact precisely if the class of morphisms that it inverts is stable under pullback. This is the case for topological localizations by definition.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
\textbf{(generation from a small set of morphisms)}

For every topological localization of $C$ at a strongly saturated class $\bar S$ there exists a [[small set]] of [[monomorphism in an (∞,1)-category|monomorphisms]] that generates $\bar S$.

\end{prop}
This is [[Higher Topos Theory|HTT, prop. 6.2.1.5]].

\begin{ucorollary}
Every topological localization of $C$ is necessarily [[accessible (∞,1)-category|accessible]] and exact.

\end{ucorollary}
This is [[Higher Topos Theory|HTT, cor. 6.2.1.6]]

The following proposition asserts that for the construction of [[(n,1)-topos]]es the notion of topological localization is empty: if colimits commute with products, then already every localization is topological. Accordingly, also the notion of [[hypercompletion]] is relevant only for [[(∞,1)-topos]]es.

\begin{uprop}
\textbf{(localizations of presentable $n$-categories are topological)}

Let $C$ be a [[locally presentable (∞,1)-category|locall presentable]] [[(n,1)-category]] for $n \in \mathbb{N}$ finite with [[universal colimits]]. Then every left exact [[localization of an (∞,1)-category|localization of]] $C$ is a topological localization

\end{uprop}
This is [[Higher Topos Theory|HTT, prop. 6.4.3.9]].

\begin{proof}
\ldots{}

\end{proof}
This means that every [[(n,1)-topos]] of $n$-sheaves is a localization at [[Cech nerve]]s of covers.

\textbf{Remark} Notice in this context the statement found for instance in

\begin{itemize}%
\item [[Daniel Dugger]], [[Sharon Hollander]], [[Daniel Isaksen]], \emph{Hypercovers and simplicial presheaves} (\href{http://www.math.uiuc.edu/K-theory/0563/}{web})

\end{itemize}
that a [[simplicial presheaf]] that satisfies [[descent]] on all [[Cech cover]]s already satisfies descent for all \emph{bounded} hypercovers. If the simplicial presheaf is $n$-truncated for some $n$, then it won't ``see'' $k$-bounded hypercovers for large enough $k$ anyway, and hence it follows that truncated simplicial presheaves that satisfy Cech descent already satisfy hyperdescent.

This is in line with the above statement that for $n$-toposes with finite $n$ there is no distinction between Cech descent and hyperdescent. The distinction becomes visible only for untruncated $\infty$-presheaves.

\hypertarget{for_presheaf_categories}{}\subsubsection*{{For $(\infty,1)$-presheaf $(\infty,1)$-categories}}\label{for_presheaf_categories}

Let throughout $C$ be a [[small (∞,1)-category]] and write $PSh_{(\infty,1)}(C)$ for the [[(∞,1)-category of (∞,1)-presheaves]] on $C$.

\begin{uprop}
\textbf{sheaves form a topological localization}

If $C$ is endowed with a [[(∞,1)-site|Grothendieck topology]], the inclusion

\begin{displaymath}
Sh_{(\infty,1)}(C)  \hookrightarrow PSh_{(\infty,1)}(C)
\end{displaymath}
is a topological localization.

\end{uprop}
This is [[Higher Topos Theory|HTT, Prop. 6.2.2.7]].

\begin{proof}
See .

\end{proof}
\begin{uprop}
All topological localizations of $PSh_{(\infty,1)}(C)$ arise this way:

There is a bijection between [[(∞,1)-site|Grothendieck topologies]] on $C$ and equivalence classes of topological localizations of $PSh_{(\infty,1)}(C)$.

\end{uprop}
This is [[Higher Topos Theory|HTT, prop. 6.2.2.17]].

\begin{proof}
\ldots{}

\end{proof}
\hypertarget{model_category_presentation}{}\subsubsection*{{Model category presentation}}\label{model_category_presentation}

See at \emph{[[?ech model structure on simplicial sheaves]]}.

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

Topological localizations are the topic of section 6.2, from def. 6.2.1.5 on, in

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

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



\end{document}