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\begin{document}

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\section*{local isomorphism}

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

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{axioms}{Axioms}\dotfill \pageref*{axioms} \linebreak
\noindent\hyperlink{relation_to_grothendieck_topologies}{Relation to Grothendieck topologies}\dotfill \pageref*{relation_to_grothendieck_topologies} \linebreak
\noindent\hyperlink{local_epimorphisms_from_local_isomorphisms}{Local epimorphisms from local isomorphisms}\dotfill \pageref*{local_epimorphisms_from_local_isomorphisms} \linebreak
\noindent\hyperlink{local_isomorphisms_from_local_epimorphisms}{Local isomorphisms from local epimorphisms}\dotfill \pageref*{local_isomorphisms_from_local_epimorphisms} \linebreak
\noindent\hyperlink{relation_to_lawveretierney_topologies}{Relation to Lawvere-Tierney topologies}\dotfill \pageref*{relation_to_lawveretierney_topologies} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{sheafification}{Sheafification}\dotfill \pageref*{sheafification} \linebreak
\noindent\hyperlink{characterization_and_relation_to_sieves}{Characterization and relation to sieves}\dotfill \pageref*{characterization_and_relation_to_sieves} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{local isomorphism} in a [[presheaf]] category $PSh(S)$ is a morphism that becomes an [[isomorphism]] after passing to [[sheaf|sheaves]] with respect to a given [[Grothendieck topology]] on $S$.

The collection of all local isomorphisms not only determines the [[Grothendieck topology]] but is precisely the collection of morphisms that are inverted when passing to sheaves. Hence local isomorphisms serve to understand [[sheaf|sheaves]] and [[sheafification]] in terms of the passage to a [[homotopy category]] of $PSh(S)$.

This is a particular case of the notion of [[reflective factorization system]], applied to the sheafification reflector. It is discussed in more detail at [[category of sheaves]]:

In terms of the discussion at [[geometric embedding]], local isomorphisms in $PSh(S)$ are precisely the [[calculus of fractions|multiplicative system]] $W$ that is sent to isomorphisms by the [[sheafification]] functor

\begin{displaymath}
\bar{(-)} : PSh(S) \to Sh(S)
\end{displaymath}
which is left [[exact functor|exact]] [[left adjoint]] to the [[full and faithful functor|full and faithful]] inclusion

\begin{displaymath}
Sh(S) \hookrightarrow PSh(S)
  \,.
\end{displaymath}
\hypertarget{axioms}{}\subsection*{{Axioms}}\label{axioms}

A system of \textbf{local isomorphism}s on $PSh(S)$ is any collection of morphisms satisfying

\begin{enumerate}%
\item local isomorphisms are a system of [[category with weak equivalences|weak equivalences]] (i.e. every [[isomorphism]] is a local isomorphism and they satisfy [[2-out-of-3]]);


\item a morphism $Y\to X$ is a local isomorphism if and only if its [[pullback]]

\begin{displaymath}
\itexarray{
 U \times_X Y &\to& Y
 \\
 {}^{\mathllap{loc iso}}\downarrow
 &&
 \downarrow^{\mathrlap{\Leftrightarrow loc iso}}
 \\
 U &\to& X
}
\end{displaymath}
along any morphism $U \to X$, where $U$ is [[representable presheaf|representable]], is a local isomorphism.



\end{enumerate}
\hypertarget{relation_to_grothendieck_topologies}{}\subsection*{{Relation to Grothendieck topologies}}\label{relation_to_grothendieck_topologies}

Systems of local isomorphisms on $PSh(S)$ are equivalent to [[Grothendieck topology|Grothendieck topologies]] on $S$.

The following indicates how choices of systems of local isomorphisms are equivalent to choices of systems of [[local epimorphisms]]. The claim follows by the discussion at [[local epimorphism]].

\hypertarget{local_epimorphisms_from_local_isomorphisms}{}\subsubsection*{{Local epimorphisms from local isomorphisms}}\label{local_epimorphisms_from_local_isomorphisms}

A system of [[local epimorphisms]] is defined from a system of local isomorphisms by declaring that $f : Y \to X$ is a [[local epimorphism]] precisely if $im(f) \to X$ is a local isomorphism.

\hypertarget{local_isomorphisms_from_local_epimorphisms}{}\subsubsection*{{Local isomorphisms from local epimorphisms}}\label{local_isomorphisms_from_local_epimorphisms}

Given a [[Grothendieck topology]] in terms of a system of [[local epimorphisms]], a system of local isomorphisms is constructed as follows.

A \textbf{local monomorphism} with respect to this topology is a morphism $f : A \to B$ in $[S^{op}, Set]$ such that the canonical morphism $A \to A \times_B A$ is a [[local epimorphism]].

A \textbf{local isomorphism} with respect to a Grothendieck topology is a morphism in $[S^{op}, Set]$ that is both a [[local epimorphism]] as well as a local monomorphism in the above sense.

\hypertarget{relation_to_lawveretierney_topologies}{}\subsection*{{Relation to Lawvere-Tierney topologies}}\label{relation_to_lawveretierney_topologies}

Recall that [[Grothendieck topology|Grothendieck topologies]] on a [[small category]] $S$ are in bijection with [[Lawvere-Tierney topology|Lawvere-Tierney-topologies]] on $PSh(S)$ and that [[sheafification]] with respect to a [[Lawvere-Tierney topology]] is encoded in terms of monomorphisms in $PSh(S)$ which are \emph{[[dense monomorphism|dense]]} with respect to the [[Lawvere-Tierney topology]].

We have:

the [[dense monomorphisms]] are precisely the local isomorphisms which are also ordinary [[monomorphisms]].

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

\begin{itemize}%
\item Local isomorphisms admit a left saturated [[calculus of fractions]].

\end{itemize}
\hypertarget{sheafification}{}\subsubsection*{{Sheafification}}\label{sheafification}

The [[sheafification]] functor which sends a [[presheaf]] $F$ to its weakly equivalent [[sheaf]] $\bar F$ can be realized using a [[colimit]] over local isomorphisms. See there.

\hypertarget{characterization_and_relation_to_sieves}{}\subsubsection*{{Characterization and relation to sieves}}\label{characterization_and_relation_to_sieves}

Often one concentrates on the local isomorphisms whose codomain is a [[representable functor|representable presheaf]], i.e. those of the form

\begin{displaymath}
A \to Y(U)
  \,,
\end{displaymath}
where $U$ is an object in $S$ and $Y$ is the [[Yoneda embedding]]. These come from covering [[sieves]] of a [[Grothendieck topology]] on $S$: for $U \in S$ and $\{V_i \to U\}_i$ a covering [[sieve]] on $U$, the coresponding local isomorphism is the presheaf which is the [[image]] of the joint injection map

\begin{displaymath}
\sqcup_i Y(V_i) \to Y(U)
  \,.
\end{displaymath}
Using the fact that morphisms in a presheaf category are [[strict morphisms]], so that [[image]] and [[coimage]] coincide, it is useful, with an eye towards generalizations from [[sheaves]] to [[stacks]] and [[∞-stacks]] (see in particular [[descent for simplicial presheaves]]), to say this equivalently in terms of the [[coimage]]: the local isomorphism corresponding to the covering [[sieve]] $\{V_i \to U\}$ is

\begin{displaymath}
colim (
     (\sqcup_i Y(V_i))\times_{Y(U)}
     (\sqcup_i Y(V_i))
     \stackrel{\to}{\to}
     (\sqcup_i Y(V_i))
  )
  \to 
  Y(U)
\end{displaymath}
Notice that in general these are not \emph{all} the local isomorphism with representable codomain (more generally these are [[hypercovers]], where $(\sqcup_i Y(V_i))\times_{Y(U)} (\sqcup_i Y(V_i))$ is replaced in turn by one of its covers).

(\ldots{})

Notice that local isomorphism with codomain a representable already induce general local isomorphisms using the fact that every presheaf is a colimit of representables (the [[co-Yoneda lemma]]) and that local isomorphisms/sieves are stable under [[pullback]]:

\begin{uprop}
If $A \in PSh(S)$ is a [[local object]] with respect to local isomorphisms whose codomain is a representable, then every morphism $X \to Y$ of presheaves such that for every representble $U$ and every morphism $U \to Y$ the pullback $X \times_Y U  \to U$ is a local isomorphism, the canonical morphism

\begin{displaymath}
Hom(Y,A) \to Hom(X,A)
\end{displaymath}
is an isomorphism.

\end{uprop}
\begin{proof}
We may first rewrite trivially

\begin{displaymath}
X \simeq X \times_Y Y
\end{displaymath}
and then use the [[co-Yoneda lemma]] to write (suppressing notationally the Yoneda embedding)

\begin{displaymath}
Y \simeq colim_{U \to Y} U
\end{displaymath}
and hence rewrite $(X \to Y)$ as

\begin{displaymath}
X \times_Y (\colim_{U \to Y} U) \to colim_{U \to Y} U
  \,.
\end{displaymath}
Then using that colimits of presheaves are [[commutativity of limits and colimits|stable under base change]] this is

\begin{displaymath}
(\colim_{U \to Y}(X \times_Y U)) \to colim_{U \to Y} U
  \,.
\end{displaymath}
Recall that by assumption the components $X \times_Y U \to U$ of this are local isomorphisms. Hence

\begin{displaymath}
(Hom(Y,A) \to Hom(X,A))
  =
  lim_{U \to Y}
  Hom(U, A) \to 
  \lim_{U  \to Y}
  Hom(X \times_Y U, A)
\end{displaymath}
is a limit over isomorphisms, hence an isomorphism.

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

This is in section 16.2 of

\begin{itemize}%
\item Kashiwara-Schapira, \emph{[[Categories and Sheaves]]} .

\end{itemize}
See in particular exercise 16.5 there for the characterization of [[Grothendieck topology|Grothendieck topologies]] in terms of local isomorphisms.

[[!redirects local isomorphisms]]



\end{document}