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\begin{document}

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\section*{category of sheaves}

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

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

[[!include topos theory - contents]]

\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{equivalent_characterizations}{Equivalent characterizations}\dotfill \pageref*{equivalent_characterizations} \linebreak
\noindent\hyperlink{as_localizations}{As localizations}\dotfill \pageref*{as_localizations} \linebreak
\noindent\hyperlink{AsToposes}{As toposes}\dotfill \pageref*{AsToposes} \linebreak
\noindent\hyperlink{AsAccessibleReflectiveSubcategories}{As accessible reflective subcategories}\dotfill \pageref*{AsAccessibleReflectiveSubcategories} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{dependence_on_the_site}{Dependence on the site}\dotfill \pageref*{dependence_on_the_site} \linebreak
\noindent\hyperlink{EpiMonoIsomorphisms}{Epi- mono- and isomorphisms}\dotfill \pageref*{EpiMonoIsomorphisms} \linebreak
\noindent\hyperlink{exactness_properties}{Exactness properties}\dotfill \pageref*{exactness_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}

This entry is about the properties and the characterization of the category $Sh(S)$ of ([[set]]-valued) [[sheaf|sheaves]] on a ([[small category|small]]) [[site]] $S$, which is a [[Grothendieck topos]]. Among other things it gives a definition and a characterization of the notion of [[sheaf]] itself, but for more details on [[sheaf|sheaves]] themselves see there.

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

Let $(C,J)$ be a [[site]]: a (small) [[category]] equipped with a [[coverage]].

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{category of sheaves} on $(C,J)$ is the [[full subcategory]] of the [[category of presheaves]]

\begin{displaymath}
i : Sh_J(C) \hookrightarrow
  PSh(C)
\end{displaymath}
on those presheaves which are [[sheaves]] with respect to $J$.

\end{defn}
\begin{prop}
\label{SheafInclusionIsReflective}\hypertarget{SheafInclusionIsReflective}{}
Every category of sheaves is a [[reflective subcategory]]

\begin{displaymath}
(L \dashv i) : Sh_J(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow}
  Psh(C)
  \,,
\end{displaymath}
hence a [[subtopos]] of the [[presheaf topos]]. Moreover, every such subtopos arises in this way: there is a [[bijection]] between [[Grothendieck topologies]] on $C$ and equivalence classes of [[geometric embeddings]] into $PSh(C)$.

\end{prop}
This appears for instance as (\hyperlink{Johnstone}{Johnstone, corollary C.2.1.11}). See also [[Lawvere-Tierney topology]].

\begin{proof}
Details on the first statement are at [[sheafification]]. A full proof for the second statement is at [[(∞,1)-category of (∞,1)-sheaves]] (there proven in [[(∞,1)-category theory]], but the proof is verbatim the same in [[category theory]]).

\end{proof}
\hypertarget{equivalent_characterizations}{}\subsection*{{Equivalent characterizations}}\label{equivalent_characterizations}

\hypertarget{as_localizations}{}\subsubsection*{{As localizations}}\label{as_localizations}

\begin{prop}
\label{}\hypertarget{}{}
The category of sheaves is equivalent to the [[homotopy category]] of the [[category with weak equivalences]] $PSh(C)$ with the weak equivalences given by $W =$[[local isomorphisms]]

\begin{displaymath}
Sh(S) \simeq Ho_{PSh(S)} = PSh(C)[local isomorphisms]^{-1}
  \,.
\end{displaymath}
The converse is also true: for every [[left exact functor]] $L : PSh(S) \to PSh(S)$ (preserving finite limits) which is [[adjoint functor|left adjoint]] to the inclusion of its image, there is a Grothendieck topology on $S$ such that the image of $L$ is the category of sheaves on $S$ with respect to that topology.

\end{prop}
We spell out proofs of some of the above claims.

Let $C$ be a [[small category]] and

\begin{displaymath}
i : \mathcal{E} \hookrightarrow [C^{op}, Set]
\end{displaymath}
a [[reflective subcategory]], hence a [[full subcategory]] with a [[left adjoint]] $L : [C^{op}, Set] \to \mathcal{E}$, such that moreover $L$ preserves [[finite limit]]s.

Write $W := L^{-1}(isos) \subset Mor([C^{op}, Set])$ for the [[class]] of [[morphism]]s in $[C^{op}, Set]$ that are sent to [[isomorphism]]s by $L$.

\begin{prop}
\label{CharacterizationOfLocalObjects}\hypertarget{CharacterizationOfLocalObjects}{}
A [[presheaf]] $A \in [C^{op}, Set]$ is in $\mathcal{E}$ (meaning: in the [[essential image]] of $i$) precisely if for all $f : X \to Y$ in $W$ the induced function

\begin{displaymath}
Hom(f,A) : Hom(Y,A) \to Hom(X,A)
\end{displaymath}
is a [[bijection]].

\end{prop}
\begin{proof}
If $A \simeq i \hat A$, then by the $(L \dashv i)$-[[adjunction]] [[isomorphism]] we have

\begin{displaymath}
Hom(f, i \hat A) \simeq \mathcal{E}(L(f), A)
  \,.
\end{displaymath}
But by assumption $L(f)$ is an [[isomorphism]], so the claim is immediate.

Conversely, if for all $f$ the function $Hom(f,A)$ is a bijection, define $\hat A := L(A)$ and let $\epsilon_A : A \to i L(A)$ be the $(L \dashv i)$-[[unit of an adjunction|unit]].

By the assumption that $i$ is a [[full and faithful functor]] and basic properties of [[adjoint functor]]s we habe that the counit

\begin{displaymath}
L i \to Id
\end{displaymath}
is a [[natural isomorphism]]. By the [[zig-zag law]] the composite

\begin{displaymath}
L A \stackrel{L \epsilon_A}{\to} L i L A \stackrel{\simeq}{\to} L A
\end{displaymath}
is the [[identity]] and therefore $L \epsilon$ is an [[isomorphism]] and so $\epsilon_A$ is in $W$, under our assumption on $A$.

Using this it follows that

\begin{displaymath}
Hom(\epsilon_A, A) : Hom(i L A, A) \stackrel{\simeq}{\to} Hom(A,A)
\end{displaymath}
is an [[isomorphism]]. Write $k_A : i L A \to A$ for the preimage of $id_A$ under this isomorphism, which is therefore a [[left inverse]] of $\epsilon_A$. This immediately implies that also $k_A$ is in $W$, and so we can enter the same argument with $k_A$ to find that it has a left inverse itself. But this means that $k_A$ is in fact an [[isomorphism]] and hence so is $\epsilon A$, which thus exhibits $A$ as being in the essential image of $i$.

\end{proof}
\begin{prop}
\label{WeakEquivalencesDetectedOnRepresentableCodomains}\hypertarget{WeakEquivalencesDetectedOnRepresentableCodomains}{}
A [[morphism]] $f : X \to Y$ is in $W$ precisely if for every morphism $z : j(c) \to Y$ with [[representable functor|representable]] [[domain]], the [[pullback]] $z^* f$ in

\begin{displaymath}
\itexarray{
     X \times_Y j(c) &\to& X
     \\  
     {}^{\mathllap{z^* f}}\downarrow && \downarrow^{\mathrlap{f}}
     \\
     j(c) &\stackrel{z}{\to}& Y
  }
\end{displaymath}
is in $W$.

\end{prop}
\begin{proof}
Assume first that $f$ is in $W$. Since by assumption $L$ preserves finite limits, it follows that

\begin{displaymath}
\itexarray{
     L(X \times_Y j(c)) &\to& L X
     \\  
     {}^{\mathllap{L(z^* f)}}\downarrow && \downarrow^{\mathrlap{L f}}
     \\
     L(j(c)) &\stackrel{L z}{\to}& L Y
  }
\end{displaymath}
is still a [[pullback]] diagram in $\mathcal{E}$ and hence that $L(z^* f)$ is the pullback of the [[isomorphism]] $L f$ and thus itself an isomorphism. Therefore $z^* f$ is in $W$.

Conversely, suppose that all these pullbacks are in $W$. Then use the ``[[co-Yoneda lemma]]'' to write the presheaf $Y$ as a [[colimit]] over all representables mapping into it

\begin{displaymath}
{\lim_\to}_{j(c_i) \stackrel{z_i}{\to} Y} j(c)  \stackrel{\simeq}{\to} Y
  \,.
\end{displaymath}
Forming the [[pullback]] along $f$, using that in a [[topos]] (such as our [[presheaf topos]]) colimits are preserved by pullbacks, we get

\begin{displaymath}
\itexarray{
    {\lim_\to}_i f^* j(c_i) &\stackrel{\simeq}{\to}& X
    \\   
    {}^{\mathllap{{\lim_\to}_i z_i^* f}}\downarrow && \downarrow^{\mathrlap{f}}
    \\   
    {\lim_\to}_i j(c_i)
    &\stackrel{\simeq}{\to}&
    Y
  }
  \,.
\end{displaymath}
Since $L$ preserves all colimits and finite limits, we also get

\begin{displaymath}
\itexarray{
    {\lim_\to}_i L(f^* j(c_i)) &\stackrel{\simeq}{\to}& L(X)
    \\   
    {}^{\mathllap{{\lim_\to}_i L(z_i^* f)}}\downarrow && \downarrow^{\mathrlap{L(f)}}
    \\   
    {\lim_\to}_i L(j(c_i))
    &\stackrel{\simeq}{\to}&
    L(Y)
  }
  \,.
\end{displaymath}
Since by assumption now all $L(z_i^* f )$ are isomorphisms, also ${\lim_\to}_i L(z_i^* f)$ is an isomorphism and hence three sides of the above square are isomorphisms. Therefore also $L(f)$ is and hence $f$ is in $W$.

\end{proof}
\begin{prop}
\label{LocalizationInducesGrothendieckTopology}\hypertarget{LocalizationInducesGrothendieckTopology}{}
The collection of [[sieve]]s in $W$, hence the collection of [[monomorphism]]s in $W$ whose [[codomain]] is a [[representable functor|representable]], constitute a [[Grothendieck topology]] on $C$.

\end{prop}
\begin{proof}
We check the list of axioms, given at [[Grothendieck topology]]:

\begin{enumerate}%
\item \emph{Pullbacks of covering sieves are covering} :

First of all, the pullback of a sieve along a morphism of representables is still a sieve, because [[monomorphism]]s are (as discussed there) stable under pullback.

Next, since $L$ preserves [[finite limit]]s, $L$ applied to the pullback sieve is the pullback of an isomorphism, hence is an isomorphism, hence the pullback sieve is in $W$.


\item \emph{The maximal sieve is covering.} Clear: $L$ applied to an [[isomorphism]] is an [[isomorphism]].


\item \emph{Two sieves cover precisely if their intersection covers.} This is again due to the [[pullback]]-stability of elements of $W$, due to the preservation of finite limits by $L$.


\item \emph{If all pullbacks of a sieve along morphisms of a covering sieve are covering, then the original sieve was covering} .

This is the same argument as in the second part of the proof of prop. \ref{WeakEquivalencesDetectedOnRepresentableCodomains}.



\end{enumerate}
\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
$\mathcal{E}$ is a [[Grothendieck topos]].

\end{prop}
\begin{proof}
By prop. \ref{LocalizationInducesGrothendieckTopology} and prop. \ref{CharacterizationOfLocalObjects} we are reduced to showing that an object $A$ is in $\mathcal{E}$ already if for all \emph{[[monomorphisms]]} $f$ in $W$ the function $Hom(f,A)$ is a bijection.

(\ldots{})

\end{proof}
\hypertarget{AsToposes}{}\subsubsection*{{As toposes}}\label{AsToposes}

Categories of sheaves are examples of [[categories]] that are [[topos]]es: they are the [[Grothendieck topos]]es characterized among all toposes as those satisfying [[Grothendieck topos|Giraud's axioms]].

\begin{prop}
\label{}\hypertarget{}{}
[[Grothendieck topos|Sheaf toposes]] are equivalently the [[geometric embedding|subtoposes]] of [[presheaf toposes]].

\end{prop}
This appears for instance as (\hyperlink{Johnstone}{Johnstone, corollary C.2.1.11}).

\hypertarget{AsAccessibleReflectiveSubcategories}{}\subsubsection*{{As accessible reflective subcategories}}\label{AsAccessibleReflectiveSubcategories}

\begin{prop}
\label{}\hypertarget{}{}
[[sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes]]

\end{prop}
See also at \emph{[[reflective sub-(∞,1)-category]]}.

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

\hypertarget{dependence_on_the_site}{}\subsubsection*{{Dependence on the site}}\label{dependence_on_the_site}

\begin{prop}
\label{}\hypertarget{}{}
For $(L \dashv i) : \mathcal{E} \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow}$ a category of sheaves and $(C,J)$ a [[site]] such that we have an equivalence of categories $\mathcal{E} \simeq Sh_J(C)$ we say that $C$ is a \textbf{site of definition} for the [[topos]] $\mathcal{E}$.

\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
There are always different [[site]]s $(C,J)$ whose categories of sheaves are [[equivalence of categories|equivalent]]. First of all for fixed $C$ and given a [[coverage]] $J$, the category of sheaves depends only on the [[Grothendieck topology]] generated by $J$. But there may be site structures also on inequivalent categories $C$ that have equivalent categories of sheaves.

\end{remark}
\begin{defn}
\label{InducedCoverage}\hypertarget{InducedCoverage}{}
For $(C,J)$ a [[site]] with [[coverage]] $J$ and $D \to C$ any [[subcategory]], the \textbf{induced coverage} $J_D$ on $D$ has as [[covering]] [[sieve]]s the intersections of the covering sieves of $C$ with the morphisms in $D$.

\end{defn}
\begin{defn}
\label{DenseSubSite}\hypertarget{DenseSubSite}{}
Let $(C,J)$ be a [[site]] (possibly [[large site|large]]). A [[subcategory]] $D \to C$ (not necessarily full) is called a \textbf{[[dense sub-site]]} with the \hyperlink{InducedCoverage}{induced coverage} $J_D$ if

\begin{enumerate}%
\item every object $U \in C$ has a [[covering]] $\{U_i \to U\}$ in $J$ with all $U_i$ in $D$;


\item for every morphism $f : U \to d$ in $C$ with $d \in D$ there is a [[covering]] family $\{f_i : U_i \to U\}$ such that the composites $f \circ f_i$ are in $D$.



\end{enumerate}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
If $D$ is a [[full subcategory]] then the second condition is automatic.

\end{remark}
\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(comparison lemma)}

Let $(C,J)$ be a (possibly [[large site|large]]) [[site]] with $C$ a [[locally small category]] and let $f : D \to C$ be a [[small category|small]] \hyperlink{DenseSubSite}{dense sub-site}. The pair of [[adjoint functor]]s

\begin{displaymath}
(f^* \dashv f_*) 
    : 
  PSh(D)
    \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}  
   PSh(C)
\end{displaymath}
with $f^*$ given by precomposition with $f$ and $f_*$ given by right [[Kan extension]] induces an [[equivalence of categories]] between the categories of sheaves

\begin{displaymath}
(f_* \dashv f^*) : 
  Sh_{J_D}(D)
  \underoverset
    {\underset{f_*}{\to}}{\overset{f^*}
    {\leftarrow}}
    {\simeq}
  Sh_J{C}
  \,.
\end{displaymath}
\end{theorem}
This appears as (\hyperlink{Johnstone}{Johnstone, theorm C2.2.3}).

\begin{example}
\label{}\hypertarget{}{}
\begin{itemize}%
\item Let $X$ be a [[locale]] with [[frame]] $Op(X)$ regarded as a site with the canonical coverage ($\{U_i \to U\}$ covers if the [[join]] of the $U_i$ is $U$). Let $bOp(X)$ be a [[basis for the topology]] of $X$: a complete join-[[semilattice]] such that every object of $Op(X)$ is the [[join]] of objects of $bOp(X)$. Then $bOp(X)$ is a dense sub-site.

\begin{itemize}%
\item For $X$ a [[locally contractible space]], $Op(X)$ its [[category of open subsets]] and $cOp(X)$ the full subcategory of [[contractible]] open subsets, we have that $cOp(X)$ is a dense sub-site.

\end{itemize}

\item For $C = TopManifold$ the category of all [[paracompact topological space|paracompact]] [[topological manifold]]s equipped with the [[open cover]] coverage, the category [[CartSp]]${}_{top}$ is a dense sub-site: every paracompact manifold has a [[good open cover]] by [[open balls]] [[homeomorphic]] to a [[Cartesian space]].


\item Similaryl for $C =$ [[Diff]] the category of [[smooth manifold]]s equipped with the [[good open cover]] [[coverage]], the full subcategory [[CartSp]]${}_{smooth}$ is a dense sub-site.



\end{itemize}
\end{example}
\hypertarget{EpiMonoIsomorphisms}{}\subsubsection*{{Epi- mono- and isomorphisms}}\label{EpiMonoIsomorphisms}

\begin{prop}
\label{RecognitionOfEpimorphisms}\hypertarget{RecognitionOfEpimorphisms}{}
Let $\mathcal{C}$ be a [[small site]] and let $Sh(\mathcal{C})$ be its category of sheaves. Let $f \colon X \to Y$ be a homomorphism of sheaves, hence a morphism in $Sh(\mathcal{C})$. Then:

\begin{enumerate}%
\item $f$ is a [[monomorphism]] or [[isomorphism]] precisely if it is so \emph{globally} in that for each object $U \in \mathcal{C}$ in the site, then the component $f_U \colon X(U) \to Y(U)$ is an [[injection]] or [[bijection]] of [[sets]], respectively.


\item $f$ is an [[epimorphism]] precisely if it is so \emph{locally}, in that: for all $U \in C$ there is a [[covering]] $\{p_i : U_i \to U\}_{i \in I}$ such that for all $i \in I$ and every element $y \in Y(U)$ the element $f(p_i)(y)$ is in the image of $f(U_i) : X(U_i) \to Y(U_i)$.



\end{enumerate}
But if $\{x_i\}_{i \in I}$ is a [[set]] of [[points of a topos]] for $Sh(\mathcal{C})$ such that these are \emph{enough points} (\href{point+of+a+topos#EnoughPoints}{def.}) then the morphism $f$ is epi/mono/iso precisely it is is so an all [[stalks]], hence precisely if

\begin{displaymath}
x_i^\ast f \;\colon\; x_i^\ast X \longrightarrow x_i^\ast Y
\end{displaymath}
is a surjection/injection/bijection of sets, respectively, for all $i \in I$.

\end{prop}
\hypertarget{exactness_properties}{}\subsubsection*{{Exactness properties}}\label{exactness_properties}

Every sheaf topos satisfies the following [[exactness properties]]. it is an

\begin{itemize}%
\item [[extensive category]];


\item [[adhesive category]];


\item [[exhaustive category]].



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

\begin{itemize}%
\item [[coverage]]

[[trivial coverage]]


\item \textbf{category of sheaves}


\item [[category of sheaves on a topological space]]


\item [[(∞,1)-category of (∞,1)-sheaves]].



\end{itemize}
[[!include locally presentable categories - table]]

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

\begin{itemize}%
\item [[Francis Borceux]], vol 3 of \emph{[[Handbook of Categorical Algebra]]}, Cambridge University Press (1994)


\item [[Peter Johnstone]], sections A.4 and C.2 in \emph{[[Elephant|Sketches of an Elephant]]}


\item [[The Stacks Project]], \emph{Sites and sheaves} (\href{http://stacks.math.columbia.edu/download/sites.pdf}{pdf})



\end{itemize}
The characterization of sheaf toposes and Grothendieck topologies in terms of left exact [[reflective subcategory|reflective subcategories]] of a presheaf category is also in

\begin{itemize}%
\item [[Saunders MacLane]], [[Ieke Moerdijk]], \emph{[[Sheaves in Geometry and Logic]]}

\end{itemize}
where it is implied by the combination of Corollary VII 4.7 and theorem V.4.1.

The characterization of $Sh(S)$ as the [[homotopy category]] of $PSh(S)$ with respect to local isomorphisms is emphasized at the beginning of the text

\begin{itemize}%
\item [[Bertrand Toen]], \emph{Stacks and non-abelian cohomology} (\href{http://www.msri.org/publications/ln/msri/2002/introstacks/toen/1/index.html}{web}) .

\end{itemize}
Details are in

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

\end{itemize}
It's a bit odd that the full final statement does not seem to be stated there explicitly, but all the ingredients are discussed:

\begin{itemize}%
\item exercise 16.7 shows that [[sheafification]] inverts precisely the [[local isomorphism]]s, so that in particular every [[local isomorphism]] between [[sheaf|sheaves]] is an [[isomorphism]];


\item lemma 16.3.2 states that the unit of the [[adjoint functor|adjunction]] $Id_{PSh(S)} \rightarrow i \circ \bar{(-)} : PSh(S) \to PSh(S)$ is componentwise a [[local isomorphism]];


\item using this corollary 7.2.2 says that $Sh(S) \simeq Ho_{PSh(S)}$ with the [[homotopy category]] $Ho_{PSh(S)}$ formed using [[local isomorphism]]s as weak equivalences.



\end{itemize}
[[!redirects sheaf category]] [[!redirects categories of sheaves]] [[!redirects topos of sheaves]] [[!redirects topoi of sheaves]] [[!redirects toposes of sheaves]]



\end{document}