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\begin{document}

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\section*{sheaf}

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

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

[[!include descent and locality - contents]]

\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{GeneralDefinitionInComponents}{General definition in components}\dotfill \pageref*{GeneralDefinitionInComponents} \linebreak
\noindent\hyperlink{GeneralDefinitionAbstractly}{General definition abstractly}\dotfill \pageref*{GeneralDefinitionAbstractly} \linebreak
\noindent\hyperlink{characterizations_over_special_sites}{Characterizations over special sites}\dotfill \pageref*{characterizations_over_special_sites} \linebreak
\noindent\hyperlink{CharacterizationsOverSitesOfOpens}{Characterizations over sites of opens}\dotfill \pageref*{CharacterizationsOverSitesOfOpens} \linebreak
\noindent\hyperlink{CharacterizationOverCanonicalTopologies}{Characterization over canonical topologies}\dotfill \pageref*{CharacterizationOverCanonicalTopologies} \linebreak
\noindent\hyperlink{sheaves_and_localization}{Sheaves and localization}\dotfill \pageref*{sheaves_and_localization} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

A [[presheaf]] on a [[site]] is a \emph{sheaf} if its value on any object of the site is given by its compatible values on any [[covering]] of that object.

See also

\begin{itemize}%
\item [[motivation for sheaves, cohomology and higher stacks]].

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

There are several equivalent ways to characterize sheaves. We start with the general but explicit componentwise definition and then discuss more [[category theory|general abstract]] equivalent reformulations. Finally we give special discussion applicable in various common special cases.

\hypertarget{GeneralDefinitionInComponents}{}\subsubsection*{{General definition in components}}\label{GeneralDefinitionInComponents}

The following is an explicit component-wise definition of sheaves that is fully general (for instance not assuming that the [[site]] has [[pullback]]s).

\begin{defn}
\label{GeneralComponentwiseDefinition}\hypertarget{GeneralComponentwiseDefinition}{}
Let $(C,J)$ be a [[site]] in the form of a [[small category]] $C$ equipped with a [[coverage]] $J$.

A [[presheaf]] $A \in PSh(C)$ is a \textbf{sheaf} with respect to $J$ if

\begin{itemize}%
\item for every [[covering]] family $\{p_i : U_i \to U\}_{i \in I}$ in $J$


\item and for every \emph{[[matching family|compatible family]] of elements}, given by tuples $(s_i \in A(U_i))_{i \in I}$ such that for all $j,k \in I$ and all [[morphisms]] $U_j \stackrel{f}{\leftarrow} K \stackrel{g}{\to} U_k$ in $C$ with $p_j \circ f = p_k \circ g$ we have $A(f)(s_j) = A(g)(s_k) \in A(K)$



\end{itemize}
then

\begin{itemize}%
\item there is a \emph{unique} element $s \in A(U)$ such that $A(p_i)(s) = s_i$ for all $i \in I$.

\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
If in the above definition there is \emph{at most} one such $s$, we say that $A$ is a [[separated presheaf]] with respect to $J$.

\end{remark}
In this form the definition appears for instance in (\hyperlink{Johnstone}{Johnstone, def. C2.1.2}).-

\hypertarget{GeneralDefinitionAbstractly}{}\subsubsection*{{General definition abstractly}}\label{GeneralDefinitionAbstractly}

We now reformulate the \hyperlink{GeneralComponentwiseDefinition}{above component-wise definition} in [[category theory|general abstract]] terms.

Write

\begin{displaymath}
j : C \hookrightarrow PSh(C)
\end{displaymath}
for the [[Yoneda embedding]].

\begin{defn}
\label{}\hypertarget{}{}
Given a [[covering]] family $\{f_i : U_i \to U\}$ in $J$, its \textbf{[[sieve]]} is the presheaf $S(\{U_i\})$ defined as the [[coequalizer]]

\begin{displaymath}
\coprod_{j,k} j(U_j) \times_{j(U)} j(U_k) 
     \stackrel{\overset{}{\to}}{\to}
   \coprod_i j(U_i) \to S(\{U_i\})
\end{displaymath}
in $PSh(C)$.

\end{defn}
Here the [[coproduct]] on the left is over the [[pullbacks]]

\begin{displaymath}
\itexarray{
    j(U_j) \times_{j(U)} j(U_k) &\stackrel{p_j}{\to}& j(U_j)
    \\
    {}^{\mathllap{p_k}}\downarrow && \downarrow^{\mathrlap{j(f_j)}}
    \\
    j(U_k) &\stackrel{j(f_k)}{\to}& j(U)
  }
\end{displaymath}
in $PSh(C)$, and the two morphisms between the coproducts are those induced componentwise by the two projections $p_j, p_k$ in this pullback diagram.

\begin{remark}
\label{}\hypertarget{}{}
Using that [[limits]] and [[colimits]] in a [[category of presheaves]] are computed objectwise, we find that the [[sieve]] $S(\{U_i\})$ defined this way is the presheaf that sends any $K \in C$ to the set of [[morphisms]] $K \to U$ in $C$ that factor through one of the $f_i$.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
For every [[covering]] family there is a canonical morphism

\begin{displaymath}
i_{\{U_i\}} :  S(\{U_i\}) \to j(U)
\end{displaymath}
that is induced by the [[universal property]] of the [[coequalizer]] from the morphisms $j(f_i) : j(U_i) \to j(U)$ and $j(U_j) \times_{j(U)} j(U_k) \to J(U)$.

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
A \textbf{sheaf} on $(C,J)$ is a [[presheaf]] $A \in PSh(C)$ that is a [[local object]] with respect to all $i_{\{U_i\}}$: an object such that for all [[covering]] families $\{f_i : U_i \to U\}$ in $J$ we have that the [[hom-functor]] $PSh_C(-,A)$ sends the canonical morphisms $i_{\{U_i\}} : S(\{U_i\}) \to j(U)$ to [[isomorphisms]].

\begin{displaymath}
PSh_C(i_{\{U_i\}}, A) : 
  PSh_C(j(U), A) \stackrel{\simeq}{\to} PSh_C(S(\{U_i\}), A)
  \,.
\end{displaymath}
\end{defn}
Equivalently, using the [[Yoneda lemma]] and the fact that the [[hom-functor]] $PSh_C(-,A)$ sends [[colimits]] to [[limits]], this says that the diagram

\begin{displaymath}
A(U) 
   \to 
  \prod_i A(U_i)
   \stackrel{\to}{\to}
  \prod_{j,k} PSh_C(j(U_j) \times_{j(U)} j(U_k), A)
\end{displaymath}
is an [[equalizer]] diagram for each covering family.

This is also called the \textbf{[[descent]] condition} for descent along the covering family.

\begin{remark}
\label{}\hypertarget{}{}
For many examples of sites that appear in practice -- but by far not for all -- it happens that the pullback presheaves $j(U_j) \times_{j(U)} \times j(U_k)$ are themselves again representable, hence that the [[pullback]] $U_j \times_U U_k$ exists already in $C$, even before passing to the [[Yoneda embedding]].

In this special case we may apply the [[Yoneda lemma]] once more to deduce

\begin{displaymath}
PSh_C(j(U_j) \times_{j(U)} j(U_k), A)
  \simeq
  A(U_j \times_U U_k)
  \,.
\end{displaymath}
Then the sheaf condition is that all diagrams

\begin{displaymath}
A(U) 
   \to 
  \prod_i A(U_i)
   \stackrel{\to}{\to}
  \prod_{j,k} A(U_j \times_U U_k)
\end{displaymath}
are [[equalizer]] [[diagram]]s.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
The condition that $PSh_C(S(\{U_i\}), A)$ is an [[isomorphism]] is equivalent to the condition that the set $A(U)$ is isomorphic to the set of [[matching families]] $(s_i \in A(U_i))$ as it appears in the \hyperlink{GeneralComponentwiseDefinition}{above component-wise definition}.

\end{prop}
\begin{proof}
We may express the set of [[natural transformation]]s $PSh_C(j(U_j) \times_{j(U)} j(U_k), A)$ (as described there) by the [[end]]

\begin{displaymath}
PSh_C(j(U_j) \times_{j(U)} j(U_k), A)
  \simeq
  \int_{K \in C}
   Set( C(K,U_j) \times_{C(K,U)} C(K,U_k) , A(K))
  \,.
\end{displaymath}
Using this in the expression of the [[equalizer]]

\begin{displaymath}
\prod_i A(U_i)
  \simeq
  \prod_i 
  \int_{K \in C}
   Set( C(K,U_i), A(K))
  \stackrel{\to}{\to}
   \prod_{j,k}
  \int_{K \in C}
   Set( C(K,U_j) \times_{C(K,U)} C(K,U_k) , A(K))
\end{displaymath}
as a [[subset]] of the product set on the left manifestly yields the componenwise definition above.

\end{proof}
\begin{defn}
\label{}\hypertarget{}{}
A \textbf{morphism of sheaves} is just a morphism of the underlying presheaves. So the [[category of sheaves]] $Sh_J(C)$ is the [[full subcategory]] of the [[category of presheaves]] on the sheaves:

\begin{displaymath}
Sh_J(C) \hookrightarrow PSh(C)
\end{displaymath}
\end{defn}
\hypertarget{characterizations_over_special_sites}{}\subsubsection*{{Characterizations over special sites}}\label{characterizations_over_special_sites}

We discuss equivalent characterizations of sheaves that are applicable if the underlying [[site]] enjoys certain special properties.

\hypertarget{CharacterizationsOverSitesOfOpens}{}\paragraph*{{Characterizations over sites of opens}}\label{CharacterizationsOverSitesOfOpens}

An important special case of sheaves is those over a [[(0,1)-site]] such as a [[category of open subsets]] $Op(X)$ of a [[topological space]] $X$. We consider some equivalent ways of characterizing sheaves among presheaves in such a situation.

(The following was mentioned in Peter LeFanu Lumsdaine's comment \href{http://mathoverflow.net/questions/23268/geometric-intuition-for-limits/23276#23276}{here}).

\begin{prop}
\label{AsContinuousFunctorOverOpenSubsets}\hypertarget{AsContinuousFunctorOverOpenSubsets}{}
Suppose $Op = Op(X)$ is the [[category of open subsets]] of some [[topological space]] regarded as a [[site]] with the canonical [[coverage]] where $\{U_i \hookrightarrow U\}$ is [[covering]] if the [[union]] $\cup_i U_i \simeq U$ in $Op$.

Then a [[presheaf]] $\mathcal{F}$ on $Op$ is a \textbf{sheaf} precisely if for every [[complete category|complete]] [[subcategory|full subcategory]] $\mathcal{U} \hookrightarrow Op$, $\mathcal{F}$ takes the [[colimit]] in $Op$ over $\mathcal{U} \hookrightarrow Op$ to a [[limit]]:

\begin{displaymath}
\mathcal{F}(\underset{\to}{lim} \mathcal{U}) \simeq \underset{\leftarrow}{lim} \mathcal{F}(\mathcal{U})
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
A complete full subcategory $\mathcal{U} \hookrightarrow Op$ is a collection $\{U_i \hookrightarrow X\}$ of [[open subsets]] that is closed under forming [[intersections]] of subsets. The [[colimit]]

\begin{displaymath}
\underset{\to}{\lim} (\mathcal{U} \hookrightarrow Op) 
  \simeq
  \cup_{i \in I} U_i
\end{displaymath}
is the [[union]] $U \coloneqq \cup_{i \in I} U_i$ of all these open subsets. Notice that by construction the component maps $\{U_i \hookrightarrow U\}$ of the colimit are a [[covering]] family of $U$.

Inspection then shows that the [[limit]] $\underset{\leftarrow}{\lim}_{i \in I} \mathcal{F}(U_i)$ is the corresponding set of [[matching families]] (use the description of \href{http://ncatlab.org/nlab/show/limit#ConstructionFromProductsAndEqualizers}{limits in terms of products and equalizers} ). Hence the statement follows with def. \ref{GeneralComponentwiseDefinition}.

\end{proof}
\hypertarget{CharacterizationOverCanonicalTopologies}{}\paragraph*{{Characterization over canonical topologies}}\label{CharacterizationOverCanonicalTopologies}

The above prop. \ref{AsContinuousFunctorOverOpenSubsets} shows that often sheaves are characterized as contravariant functors that take some [[colimits]] to [[limits]]. This is true in full generality for the following case

\begin{prop}
\label{AsContinuousFunctorsOnCanonicalTopology}\hypertarget{AsContinuousFunctorsOnCanonicalTopology}{}
Let $\mathcal{T}$ be be a [[topos]], regarded as a [[large site]] when equipped with the [[canonical topology]]. Then a [[presheaf]] (with values in [[small sets]]) on $\mathcal{T}$ is a sheaf precisely if it sends all [[colimits]] to [[limits]].

\end{prop}
\hypertarget{sheaves_and_localization}{}\subsection*{{Sheaves and localization}}\label{sheaves_and_localization}

We now describe the derivation and the detailed description of various aspects of sheaves, the [[descent]] condition for sheaves and [[sheafification]], relating it to all the related notions

\begin{itemize}%
\item [[geometric embedding]]

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


\item [[homotopy category]]



\end{itemize}

\item [[coverage]]

\begin{itemize}%
\item [[Grothendieck topology]]


\item [[Lawvere-Tierney topology]]



\end{itemize}

\item [[local isomorphism]]

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

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


\item [[hypercover]]



\end{itemize}

\item [[dense monomorphism]]


\item [[local epimorphism]]



\end{itemize}


\end{itemize}
We start by assuming that a [[geometric embedding]] into a [[presheaf]] category is given and derive the consequences.

So let $S$ be a [[small category]] and write $PSh(S) = PSh_S = [S^{op}, Set]$ for the corresponding [[topos]] of [[presheaf|presheaves]].

Assume then that another topos $Sh(S) = Sh_S$ is given together with a [[geometric embedding]]

\begin{displaymath}
f : Sh(S) \to PSh(S)
\end{displaymath}
i.e. with a [[full and faithful functor]]

\begin{displaymath}
f_* : Sh(S) \to PSh(S)
\end{displaymath}
and a left [[exact functor]]

\begin{displaymath}
f^* : PSh(S) \to Sh(S)
\end{displaymath}
Such that both form a pair of [[adjoint functor]]s

\begin{displaymath}
f^* \dashv f_*
\end{displaymath}
with $f^*$ [[left adjoint]] to $f_*$.

Write $W$ for the category

\begin{displaymath}
Core(PSh(S)) \hookrightarrow W \hookrightarrow PSh(S)
\end{displaymath}
consisting of all those morphisms in $PSh(S)$ that are sent to [[isomorphism]]s under $f^*$.

\begin{displaymath}
W = (f^*)^{-1}(Core(Sh_S))
  \,.
\end{displaymath}
From the discussion at [[geometric embedding]] we know that $Sh(S)$ is equivalent to the full [[subcategory]] of $PSh(S)$ on all $W$-[[local object]]s.

Recall that an object $A \in PSh(S)$ is called a $W$-[[local object]] if for all $p : Y \to X$ in $W$ the morphism

\begin{displaymath}
p^* : PSh_S(X,A) \to PSh_S(Y,A)
\end{displaymath}
is an [[isomorphism]]. This we call the [[descent]] condition on presheaves (saying that a presheaf ``descends'' along $p$ from $Y$ ``down to'' $X$). Our task is therefore to identify the category $W$, show how it determines and is determed by a [[Grothendieck topology]] on $S$ -- equipping $S$ with the structure of a [[site]] -- and characterize the $W$-[[local object]]s. These are (up to equivalence of categories) the objects of $Sh$, i.e. the sheaves with respect to the given [[Grothendieck topology]].

\begin{ulemma}
A morphism $Y \to X$ is in $W$ if and only if for every [[representable functor|representable presheaf]] $U$ and every morphism $U\to X$ the pullback $Y \times_X U \to U$ is in $W$

\begin{displaymath}
\itexarray{
     Y \times_X U &\to& Y
     \\
     \downarrow^{\in W} && \downarrow^{\Leftrightarrow \in W}
     \\
     U &\to& X
  }
  \,.
\end{displaymath}
\end{ulemma}
\begin{proof}
Since $W$ is stable under [[pullback]] (as described at [[geometric embedding]]: simply because $f^*$ preserves finite limits) it is clear that $Y \times_X U \to U$ is in $W$ if $Y \to X$ is.

To get the other direction, use the [[co-Yoneda lemma]] to write $X$ as a [[colimit]] of [[representable functor|representables]] over the [[comma category]] $(Y/const_X)$ (with $Y$ the [[Yoneda embedding]]):

\begin{displaymath}
X \simeq colim_{U_i \to X} U_i
  \,.
\end{displaymath}
Then pull back $Y \to colim_{U_i \to X} U$ over the entire colimiting cone, so that over each component we have

\begin{displaymath}
\itexarray{
    Y \times_X U_i &\to& Y
    \\
    \downarrow && \downarrow
    \\
    U_i &\to& X
  }
  \,.
\end{displaymath}
Using that in $PSh(S)$ [[commutativity of limits and colimits|colimits are stable under base change]] we get

\begin{displaymath}
colim_i (Y \times_X U_i) \simeq (colim_i U_i) \times_X Y
  \,.
\end{displaymath}
But since $X \simeq colim_i U_i$ the right hand is $X \times_X Y$, which is just $Y$. So $Y = colim_i (Y \times_X U_i)$ and we find that $Y \to X$ is a morphism of colimits. But under $f^*$ the two respective diagrams become isomorphic, since $Y \times_X U_i \to U_i$ is in $W$. That means that the corresponding morphism of colimits $f^*(Y \to X)$ (since $f^*$ preserves colimits) is an isomorphism, which finally means that $Y \to X$ is in $W$.

\end{proof}
\begin{ulemma}
A presheaf $A \in PSh(S)$ is a [[local object]] with respect to all of $W$ already if it is local with respect to those morphisms in $W$ whose codomain is [[representable functor|representable]]

\end{ulemma}
\begin{proof}
Rewriting the morphism $Y \to X$ in $W$ in terms of colimits as in the above proof

\begin{displaymath}
\itexarray{  
     colim_{U \to X} U_i \times_X Y 
      &\stackrel{\simeq}{\to}& Y 
     \\
     \downarrow && \downarrow
     \\
     colim_{U \to X} U &\stackrel{\simeq}{\to}& X
  }
\end{displaymath}
we find that $A(X) \to A(Y)$ equals

\begin{displaymath}
lim_{U \to X} (A(U) \to A(U \times_X Y))
  \,.
\end{displaymath}
If $A$ is local with respect to morphisms $W$ with representable codomain, then by the above if $Y \to X$ is in $W$ all the morphisms in the limit here are isomorphisms, hence

\begin{displaymath}
\cdots = Id_{A(X)}
  \,.
\end{displaymath}
\end{proof}
\begin{ulemma}
Every morphism $Y \to X$ in $W \subset PSh(S)$ factors as an epimorphism followed by a monomorphism in $PSh(S)$ with both being morphisms in $W$.

\end{ulemma}
\begin{proof}
Use factorization through [[image]] and [[coimage]], use exactness of $f^*$ to deduce that the factorization exists not only in $PSh(S)$ but even in $W$.

More in detail, given $Y \to X$ we get the diagram

\begin{displaymath}
\itexarray{
    Y \times_X Y &&\to&& Y
    \\
     &&& \swarrow
    \\
    \downarrow &&Y \sqcup_{Y \times_X Y} Y && \downarrow
    \\
    & \nearrow && \searrow
    \\
    Y && \to && X
  }
  \,.
\end{displaymath}
Because $f^*$ is exact, the pullbacks and pushouts in this diagram remain such under $f^*$. But since $f^*(Y \to X)$ is an isomorphism by assumption, the all these are pullbacks and pushouts along isomorphisms in $Sh(S)$, so all morphisms in the above diagram map to isomorphisms in $Sh(S)$, hence the entire diagram in $PSh(S)$ is in $W$.

Since the morphism $Y \sqcup_{Y \times_X Y} Y  \to X$ out of the [[coimage]] is at the same time the [[equalizer|equalizing]] morphism into the [[image]] $lim(X \stackrel{\to}{\to} X \sqcup_Y X)$, it is a [[monomorphism]].

\end{proof}
\begin{udefinition}
The monomorphisms in $PSh(S)$ which are in $W$ are called [[dense monomorphism]]s.

\end{udefinition}
\begin{ulemma}
Every [[monomorphism]] $Y \to X$ with $X$ [[representable functor|representable]] is of the form

\begin{displaymath}
Y = colim ( U \times_X U \to U  )
\end{displaymath}
for $U = \sqcup_{\alpha} U_\alpha$ a disjoint union of representables

\end{ulemma}
\begin{proof}
This is a direct consequence of the standard fact that subfunctors are in bijection with [[sieve]]s.

\end{proof}
\begin{ucorollary}
If a presheaf $A$ is [[local object|local]] with respect to all [[dense monomorphism]]s, then it is already local with respect to all morphisms $Y \to X$ of the form

\begin{displaymath}
\itexarray{
    Y
    \\
    \downarrow
    \\
    X
  }
  =
  colim
  \left(
  \itexarray{
    W &\stackrel{\to}{\to}& U 
    \\
    \;\;\downarrow^{dense mono} && \downarrow^{Id}
    \\
    U \times_X U & \stackrel{\to}{\to}&  U
  }
  \right)
\end{displaymath}
with the left vertical morphism a [[dense monomorphism]]

(and with $U = \sqcup_\alpha U_\alpha$ the disjoint union (of representable presheaves) over a [[cover]]ing family of objects.)

\end{ucorollary}
\begin{udefinition}
The morphisms in $W$ with representable codomain

\begin{itemize}%
\item of the form $colim (U \times_X U \stackrel{\to}{\to} U) \to X$ as above are [[cover]]s:


\item of the form $colim (W  \stackrel{\to}{\to} U) \to X$ (with $W$ a cover of $U \times_X U$) as above are [[hypercover]]s



\end{itemize}
of the representable $X$.

\end{udefinition}
\begin{uproposition}
A presheaf $A$ is $W$-local, i.e. a sheaf, already if it is local (satisfies [[descent]]) with respect to all [[cover]]s, i.e. all [[dense monomorphism]]s with codomain a [[representable functor|representable]].

\end{uproposition}
\begin{quote}%
[[Urs Schreiber|Urs]]: the above shows this almost. I am not sure yet how to see the remaining bit directly, without making recourse to the full machinery leading up to section VII, 4, corollary 7 in [[Sheaves in Geometry and Logic]].


\end{quote}
So we finally conclude:

\begin{ucorollary}
We have:

\begin{itemize}%
\item Systems $W$ of weak equivalences defined by choice of [[geometric embedding]] $f : Sh(S) \to PSh(S)$ are in canonical bijection with choice of [[Grothendieck topology]].


\item A presheaf $A$ is $W$-local, i.e. local with respect to all [[local isomorphism]]s, if and only if it is local already with respect to all [[dense monomorphism]], i.e. if and only if it satisfies sheaf condition for all covering [[sieve]]s.



\end{itemize}
\end{ucorollary}
From the \emph{assumption} that $f : Sh(S) \to PSh(S)$ is a [[geometric embedding]] follows at once the following explicit description of the [[sheafification]] functor $f^* : PSh(S) \to Sh(S)$.

\begin{ulemma}
For $A \in PSh(S)$ a presheaf, its [[sheafification]] $\bar A := f_* f^* A$ is the presheaf given by

\begin{displaymath}
\bar A : U \mapsto colim_{(Y \to U) \in W} A(Y)
\end{displaymath}
\end{ulemma}
\begin{proof}
By the discussion at [[geometric embedding]] the category $Sh(S)$ is equivalent to the [[localization]] $PSh(S)[W^{-1}]$, which in turn is the category with the same objects as $PSh(S)$ and with morphisms given by spans out of hypercovers in $W$

\begin{displaymath}
PSh(S)[W^{-1}](X,A) =
  colim_{(Y \to X) \in W} A(Y)
  \,.
\end{displaymath}
So we have

\begin{displaymath}
\array {
  Sh(S) &&\stackrel{\stackrel{f_*}{\to}}{\stackrel{f^*}{\leftarrow}}&
  PSh(S)
  \\
  & \searrow_{\simeq}&\Downarrow^{\simeq}&  \downarrow
  \\
  &&
  PSh(S)[W^{-1}]
  \,.
}
\end{displaymath}
and deduce

\begin{itemize}%
\item by [[Yoneda lemma|Yoneda]] that $\bar A(U) = PSh_S(U, \bar A)$;


\item by the [[adjoint functor|hom-adjunction]] this is $\cdots \simeq Sh_S(\bar U, \bar A)$;


\item by the equivalence just mentioned this is $\cdots \simeq PSh_S[W^{-1}](U,A)$.



\end{itemize}
\end{proof}
\begin{uremark}
For checking the sheaf condition the [[dense monomorphism]]s, i.e. the ordinary [[cover]]s are already sufficient. But for [[sheafification]] one really needs the [[local isomorphism]]s, i.e. the [[hypercover]]s. If one takes the colimit in the sheafification prescription above only over [[cover]]s, one obtains instead of sheafification the plus-construction.

\end{uremark}
\begin{udefinition}
For $A \in PSh(S)$ a presheaf, the \textbf{plus-construction} on $A$ is the presheaf

\begin{displaymath}
A^+ : X \mapsto  colim_{(Y \hookrightarrow X) \in W } A(Y)
\end{displaymath}
where the colimit is over all [[dense monomorphism]]s (instead of over all [[local isomorphism]]s as for [[sheafification]] $\bar A$).

\end{udefinition}
\begin{uremark}
In general $A^+$ is not yet a sheaf. It is however in general closer to being a sheaf than $A$ is, namely it is a [[separated presheaf]].

But applying the plus-construction twice yields the desired sheaf

\begin{displaymath}
(A^+)^+ = \bar A
  \,.
\end{displaymath}
This is essentially due to the fact that in the context of ordinary sheaves discussed here, all [[hypercover]]s are already of the form

\begin{displaymath}
colim(W \stackrel{\to}{\to} U)
\end{displaymath}
for $W \to U \times_X U$ a cover. For higher [[stack]]s the hypercover is in general a longer simplicial object of covers and accordingly if one restricts to covers instead of using hypercovers one will need to use the plus-construction more and more often.

\end{uremark}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

The archetypical example of sheaves are \emph{sheaves of [[function]]s}:

\begin{itemize}%
\item for $X$ a topological space, $\mathbb{C}$ a topological space and $O(X)$ the [[site]] of open subsets of $X$, the assignment $U \mapsto C(U,\mathbb{C})$ of continuous functions from $U$ to $\mathbb{C}$ for every open subset $U \subset X$ is a sheaf on $O(X)$.


\item for $X$ a complex manifold and $\mathbb{C}$ a complex manifold, the assignment $U \mapsto C_{hol}{X,\mathbb{C}}$ of holomorphic functions in a sheaf.



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

\begin{itemize}%
\item [[presheaf]] / [[separated presheaf]] / \textbf{sheaf} / [[cosheaf]]

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


\item [[abelian sheaf]], [[sheaf of abelian groups]], [[sheaf of modules]], [[quasicoherent sheaf]], [[sheaf of meromorphic functions]]


\item [[locally constant sheaf]], [[constructible sheaf]]


\item [[sheaf with transfer]]



\end{itemize}

\item [[2-sheaf]] / [[stack]]


\item [[(∞,1)-sheaf]] / [[∞-stack]]

\begin{itemize}%
\item [[sheaf of spectra]]

\end{itemize}

\item [[(∞,2)-sheaf]]


\item [[(∞,n)-sheaf]]


\item [[abelian sheaf cohomology]]

\begin{itemize}%
\item [[soft sheaf]]


\item [[fine sheaf]]


\item [[flabby sheaf]]



\end{itemize}

\item [[descent]]

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


\item [[cohomological descent]]


\item [[descent morphism]]


\item [[monadic descent]],

\begin{itemize}%
\item [[Sweedler coring]]


\item [[higher monadic descent]]


\item [[descent in noncommutative algebraic geometry]]



\end{itemize}


\end{itemize}


\end{itemize}
[[!include homotopy n-types - table]]

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

Section C2 in

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]}

\end{itemize}
\begin{itemize}%
\item [[Saunders MacLane]], [[Ieke Moerdijk]], \emph{[[Sheaves in Geometry and Logic]]}
\item [[Yuri Manin]], \emph{Methods of homological algebra}

\end{itemize}
A concise and contemporary overview can be found in

\begin{itemize}%
\item C. Centazzo, [[Enrico Vitale|E. M. Vitale]], \emph{Sheaf theory} , pp.311-358 in Pedicchio, Tholen (eds.), \emph{Categorical Foundations} , Cambridge UP 2004. (\href{https://perso.uclouvain.be/enrico.vitale/chapter7.pdf}{draft})

\end{itemize}
The book by Kashiwara and Schapira discusses sheaves with motivation from [[homological algebra]], [[abelian sheaf cohomology]] and [[homotopy theory]], leading over in the last chapter to the notion of [[stack]].

\begin{itemize}%
\item [[Masaki Kashiwara]], [[Pierre Schapira]], \emph{[[Categories and Sheaves]]}, Grundlehren der Mathematischen Wissenschaften \textbf{332}, Springer (2006)

\end{itemize}
A quick pedagogical introduction with an eye towards the generalization to [[(∞,1)-sheaves]] is in

\begin{itemize}%
\item [[Dan Dugger]], \emph{Sheaves and homotopy theory}, \href{http://ncatlab.org/nlab/files/cech.pdf}{pdf}

\end{itemize}
Classics of sheaf theory on topological spaces are

\begin{itemize}%
\item Roger Godement, \emph{Topologie alg\'e{}brique et th\'e{}orie des faisceaux}, Hermann, 1958, 283 p. \href{http://books.google.fr/books/about/Topologie_alg%C3%A9brique_et_th%C3%A9orie_des_fa.html?id=JVrvAAAAMAAJ}{gBooks}


\item [[A. Grothendieck]], [[Tohoku]]



\end{itemize}
Recently, an improvement in understanding the interplay of derived functors (inverse image and proper direct image) in sheaf theory on topological spaces has been exhibited in

\begin{itemize}%
\item Olaf M. Schnuerer, Wolfgang Sergel, \emph{Proper base change for separated locally proper maps}, \href{http://arxiv.org/abs/1404.7630}{arxiv/1404.7630}

\end{itemize}
[[!redirects sheaves]]

[[!redirects 1-sheaf]] [[!redirects (1,1)-sheaf]]

[[!redirects 1-sheaves]] [[!redirects (1,1)-sheaves]]



\end{document}