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\begin{document}

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\section*{local epimorphism}

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToSieves}{Relation to sieves}\dotfill \pageref*{RelationToSieves} \linebreak
\noindent\hyperlink{relation_to_simplicial_presheaves}{Relation to simplicial presheaves}\dotfill \pageref*{relation_to_simplicial_presheaves} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Given a [[site]] $\mathcal{S}$, then a \emph{local epimorphism} is a [[morphism]] in the [[category of presheaves]] over the site which becomes an [[epimorphism]] under [[sheafification]].

More abstractly, for $\mathcal{S}$ a [[small category]], one says axiomatically that a system of \emph{local epimorphisms} is a system of [[morphisms]] in the [[presheaf]] category $[S^{op}, Set]$ that has the closure properties expected of [[epimorphisms]] under [[composition]] and under [[pullback]].

There is then a unique [[Grothendieck topology]] on $\mathcal{S}$ that induces this system of local epimorphism, see \emph{\hyperlink{RelationToSieves}{Relation to sieves}} below.

Moreover the [[local isomorphisms]] among the local epimorphisms admit a [[calculus of fractions]] which equips the [[category of presheaves]] with the structure of a [[category with weak equivalences]]. The corresponding [[reflective localization]] is the [[category of sheaves]] on the site $\mathcal{S}$.

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

\begin{defn}
\label{SystemOfLocalEpimorphisms}\hypertarget{SystemOfLocalEpimorphisms}{}
\textbf{(system of local epimorphisms)}

Let $\mathcal{S}$ be a [[small category]]. A \emph{system of local epimorphisms} on the [[presheaf category]] $[\mathcal{S}^{op}, Set]$ is a [[class]] of [[morphisms]] satisfying the following axioms:

\textbf{LE1} every [[epimorphism]] in $[\mathcal{S}^{op}, Set]$ is a local epimorphism;

\textbf{LE2} the [[composition|composite]] of two local epimorphisms is a local epimorphism;

\textbf{LE3} if the [[composition|composite]] $A_1 \stackrel{u}{\to} A_2 \stackrel{v}{\to} A_3$ is a local epimorphism, then so is $v$;

\textbf{LE4} a morphism $u \colon A \to B$ is a local epimorphism precisely if for all $U \in \mathcal{S}$ (regarded as a [[representable presheaf]]) and morphisms $y: U \to B$, the [[pullback]] morphism $A \times_B U \to U$ is a local epimorphism.

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

\hypertarget{RelationToSieves}{}\subsubsection*{{Relation to sieves}}\label{RelationToSieves}

The specification of a system of local epimorphisms is equivalent to a system of [[Grothendieck topology|Grothendieck covering]] [[sieves]].

To see this, translate between local epimorphisms to sieves as follows.

Throughout, let $\mathcal{S}$ be a [[small category]]. Write $[\mathcal{S}^{op}, Set]$ for its [[category of presheaves]] and write

\begin{displaymath}
y \;\colon\; \mathcal{S} \longrightarrow [\mathcal{S}^{op}, Set]
\end{displaymath}
for the [[Yoneda embedding]].

\begin{defn}
\label{LocalEpimorphismFromGrothendieckTopology}\hypertarget{LocalEpimorphismFromGrothendieckTopology}{}
\textbf{(local epimorphisms from [[Grothendieck topology]])}

Let the [[small category]] $\mathcal{C}$ be equipped with a [[Grothendieck topology]].

For $U \in \mathcal{S}$ an object in the site, a [[morphism]] of [[presheaves]] into the corresponding [[representable presheaf|represented presheaf]]

\begin{displaymath}
A \overset{f}{\longrightarrow} y(U) 
  \;\;\;
  \in [\mathcal{S}^{op}, Set]
\end{displaymath}
is a \emph{local epimorphism} if the [[sieve]]

\begin{displaymath}
sieve_A \subset y(U) \in [S^{op}, Set]
\end{displaymath}
at $U$ which assigns to $V$ all morphisms from $V$ to $U$ that factor through $f$

\begin{displaymath}
sieve_f 
    \;\colon\; 
  V 
   \;\mapsto\;
  \left\{  V \overset{g}{\to} U \,\in \mathcal{S} 
      \;\Big\vert\; 
      \itexarray{
        && A
        \\
        & {}^{\mathllap{\exists}}\nearrow & \Big\downarrow{}^{f}
        \\
        y(V) & \underset{y(g)}{\longrightarrow} & y(U)
      }
  \right\}
\end{displaymath}
is a [[covering sieve]].

A general morphism of presheaves

\begin{displaymath}
A \overset{}{\longrightarrow} B \;\;\; \in [\mathcal{S}^{op}, A]
\end{displaymath}
is a \emph{local epimorphism} if for every $U \in \mathcal{S}$ and every $y(U) \to B$ the [[projection]] morphism $y(U) \times_{B} A \overset{p_1}{\to} y(V)$ out of the [[pullback]]/[[fiber product]]

\begin{displaymath}
\itexarray{
    y(U)\times_{B} A
      &\overset{}{\longrightarrow}&
    A
    \\
    {}^{\mathllap{p_1}}\Big\downarrow 
    &{}^{(pb)}& \Big\downarrow{}^{\mathrlap{f}}
    \\
    y(U) &\underset{}{\longrightarrow}& B
  }
\end{displaymath}
is a local epimorphism as above. By the [[universal property]] of the [[fiber product]], this means equivalently that

\begin{displaymath}
sieve_f 
    \;\colon\; 
  V 
   \;\mapsto\;
  \left\{  V \overset{g}{\to} U \,\in \mathcal{S} 
      \;\Big\vert\; 
      \itexarray{
        y(V) &\overset{\exists}{\longrightarrow}& A
        \\
        {}^{\mathllap{g}}\Big\downarrow & & \Big\downarrow{}^{f}
        \\
        y(U) & \underset{}{\longrightarrow} & B
      }
  \right\}
\end{displaymath}
is a [[covering sieve]].

\end{defn}
\begin{remark}
\label{InTermsOfCoverages}\hypertarget{InTermsOfCoverages}{}
\textbf{(in terms of [[coverages]])}

If instead of a [[Grothendieck topology]] we are just given a [[coverage]], then Def. \ref{LocalEpimorphismFromGrothendieckTopology} becomes:

\begin{displaymath}
A \overset{f}{\longrightarrow} B
  \;\;\; \in
  [\mathcal{S}^{op}, Set]
\end{displaymath}
is a local epimorphism, if for all $y(U) \longrightarrow B$ there is a [[covering]] $\{ V_i \overset{\iota_i}{\longrightarrow} U \}$ in the coverage, such that for each $i$ there exists a lift

\begin{displaymath}
\itexarray{
    y(V_i) &\overset{\exists}{\longrightarrow}& A
    \\
    {}^{\mathllap{\iota_i}}\Big\downarrow & & \Big\downarrow{}^{f}
    \\
    y(U) & \underset{}{\longrightarrow} & B
  }
\end{displaymath}
\end{remark}
\begin{defn}
\label{GrothendieckTopologyFromLocalEpimorphisms}\hypertarget{GrothendieckTopologyFromLocalEpimorphisms}{}
\textbf{(Grothendieck topology from local epimorphisms)}

Conversely, assume a system of local epimorphisms as in Def. \ref{SystemOfLocalEpimorphisms} is given.

Declare a [[sieve]] $F$ at $U$ to be a [[covering sieve]] precisely if the inclusion morphism $F \hookrightarrow U$ is a local epimorphism. Then this defines a [[Grothendieck topology]] encoded by the collection of local epimorphisms.

\end{defn}
\hypertarget{relation_to_simplicial_presheaves}{}\subsubsection*{{Relation to simplicial presheaves}}\label{relation_to_simplicial_presheaves}

\begin{prop}
\label{CechNerveProjectionOfLocalEpimorphismIsLocalWeakEquivalence}\hypertarget{CechNerveProjectionOfLocalEpimorphismIsLocalWeakEquivalence}{}
\textbf{([[Cech nerve]] [[projection]] of [[local epimorphism]] is [[local weak equivalence]])}

For $\mathcal{S}$ a [[site]], let

\begin{displaymath}
A \overset{f}{\longrightarrow} B
  \;\colon\;
  [\mathcal{S}^{op}, Set]
\end{displaymath}
be a local epimorphism (Def. \ref{LocalEpimorphismFromGrothendieckTopology}). Then the projection

\begin{displaymath}
C(f) \longrightarrow B
  \;\;\;\;
  \in [\mathcal{S}^{op}, sSet]
\end{displaymath}
out of the [[Cech nerve]] [[simplicial presheaf]]

\begin{displaymath}
C(f)_k \;\coloneqq\; 
  \underset{
     k \; \text{factors}
  }{
  \underbrace{
    A \times_B \cdots \times_B A
  }}
\end{displaymath}
is a [[weak equivalence]] in the projective local [[model structure on simplicial presheaves]] $[\mathcal{S}^{op}, sSet_{Qu}]_{proj,loc}$.

\end{prop}
(\hyperlink{DuggerHollanderIsaksen02}{Dugger-Hollander-Isaksen 02, corollary A.3})

$\,$

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

\begin{itemize}%
\item Kashiwara-Schapira, section 16 of \emph{[[Categories and Sheaves]]}


\item [[Saunders Mac Lane]], [[Ieke Moerdijk]], chapter III, section 8 of \emph{[[Sheaves in Geometry and Logic]], Springer 1992}


\item [[Daniel Dugger]], [[Sharon Hollander]], [[Daniel Isaksen]], \emph{Hypercovers and simplicial presheaves}, Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 136. No. 1. Cambridge University Press, 2004 (\href{https://arxiv.org/abs/math/0205027}{arXiv:math/0205027})



\end{itemize}
[[!redirects local epimorphisms]] [[!redirects local epi]] [[!redirects local epis]]

[[!redirects generalized cover]] [[!redirects generalized covers]]



\end{document}