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\begin{document}

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\section*{internal sheaf}

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

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

[[!include topos theory - contents]]

\hypertarget{internal_category_theory}{}\paragraph*{{Internal category theory}}\label{internal_category_theory}

[[!include internal infinity-categories contents]]

\hypertarget{topos_theory_2}{}\paragraph*{{$(\infty,2)$-Topos theory}}\label{topos_theory_2}

[[!include (infinity,2)-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{InTermsOf2Sheaves}{In terms of external 2-sheaves}\dotfill \pageref*{InTermsOf2Sheaves} \linebreak
\noindent\hyperlink{ExplicitDefinition}{Explicit definition}\dotfill \pageref*{ExplicitDefinition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{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}

The notions of \emph{[[presheaf]]}, \emph{[[site]]} and \emph{[[sheaf]]} can be formulated [[internalization|internal]] to any [[topos]]. The ordinary such notions are recovered by internalization into [[Set]].

More precisely, the direct internalization of these notions is into the [[universe enlargement|very large]] [[2-topos]] $\hat Sh_2(\mathcal{S},can)$ of the given ambient topos $\mathcal{S}$, since an internal presheaf is to be an $\mathcal{S}$-valued internal functor, but $\mathcal{S}$ does not quite sit inside itself. It does, however, sit inside $\hat Sh_2(\mathcal{S},can)$, incarnated as the [[2-sheaf]] $\bar \mathcal{S}$ corresponding to its [[codomain fibration]].

Therefore, regarding $\hat Sh_2(\mathcal{S}, can)$ as the [[2-category]] of [[internal categories]] in $\mathcal{S}$, an [[internal site]] in $\mathcal{S}$ is an object $\bar \mathbb{C}$ of $\hat Sh_2(\mathcal{S}, can)$ and an internal presheaf is a morphism $F : \bar \mathbb{C}^{op} \to \bar \mathcal{S}$.

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

While it is straightforward to define an [[internal site]], hence the [[domain]] of an internal (pre)sheaf, the definition of the codomain is slightly more subtle, for that needs to be a copy of the ambient [[universe]] internalized into itself. One way to naturally say this is by passing to the \emph{external} [[2-sheaves]] [[2-topos]]. This version of the definition we state in

\begin{itemize}%
\item \emph{\hyperlink{InTermsOf2Sheaves}{In terms of external 2-sheaves}}.

\end{itemize}
But the resulting notion can of course be expressed entirely in terms of data in the ambient topos. This we spell out in

\begin{itemize}%
\item \emph{\hyperlink{ExplicitDefinition}{Explicit definition}}.

\end{itemize}
\hypertarget{InTermsOf2Sheaves}{}\subsubsection*{{In terms of external 2-sheaves}}\label{InTermsOf2Sheaves}

Let $\mathcal{S}$ be a [[topos]] and let $\mathbb{C}$ be an [[internal category]] in $\mathcal{S}$:

\begin{displaymath}
\mathbb{C} = 
  \left(
    C_1 \stackrel{\overset{s}{\to}}{\underset{t}{\to}}
    C_0
  \right)
  \in \mathcal{S}
  \,.
\end{displaymath}
\begin{defn}
\label{}\hypertarget{}{}
Write $\bar \mathbb{C}$ for the [[2-sheaf]] on $\mathcal{S}$

\begin{displaymath}
\bar \mathbb{C} : \mathcal{S}^{op} \to Cat
\end{displaymath}
that is [[representable functor|represented]] by $\mathbb{C}$. More explicitly, this is the [[pseudofunctor]] which to an [[object]] $X \in \mathcal{S}$ assigns

\begin{displaymath}
\bar \mathbb{C} : X 
    \mapsto 
  \left(
    \mathcal{S}(X,C_1)
    \stackrel{\overset{\mathcal{S}(X,s)}{\to}}{\underset{\mathcal{S}(X,t)}{\to}}
    \mathcal{S}(X,C_0)
  \right)
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
Write

\begin{displaymath}
\bar \mathcal{S} : \mathcal{S}^{op} \to Cat
\end{displaymath}
for the [[2-sheaf]] that [[Grothendieck construction|classifies]] the [[codomain fibration]] of $\mathcal{S}$, the [[pseudofunctor]] which sends an object to the corresponding [[slice topos]] and morphisms to [[base change]]

\begin{displaymath}
\bar \mathcal{S} : X \mapsto \mathcal{S}_{/X}
\end{displaymath}
(also called the ``[[indexed category|self-indexing]] of $\mathcal{S}$'').

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This is the incarnation of $\mathcal{S}$ itself, regard internal to its [[2-sheaf]] [[2-topos]] $\hat Sh_{2}(\mathcal{T}, can)$.

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
An \textbf{internal presheaf} on $\mathbb{C}$ (internal to $\mathcal{T}$) is a morphism

\begin{displaymath}
F : \bar \mathbb{C}^{op} \to \bar \mathcal{S}
\end{displaymath}
of [[2-sheaves]] on $\mathcal{S}$. (Also called an ``[[indexed functor]]'' between [[indexed categories]]``.)

Suppose moreover that $\mathbb{C}$ is equipped with the structure of an [[internal site]]. Then $F$ above is an \textbf{internal sheaf} on $\mathbb{C}$ if it satisfies the evident [[descent]] condition.

A [[morphism]] of internal presheaves is simply a [[2-morphism]] in $\hat Sh_2(\mathcal{S}, can)$ (also called an ``[[indexed functor|indexed natural transformation]]''). This yields a [[category]]

\begin{displaymath}
PSh(\mathbb{C})
\end{displaymath}
of internal presheaves. Accordingly we have the [[full subcategory]]

\begin{displaymath}
Sh(\mathbb{C}, \mathcal{S})
  \hookrightarrow PSh(\mathbb{C}, \mathcal{S})
\end{displaymath}
of internal sheaves.

\end{defn}
\hypertarget{ExplicitDefinition}{}\subsubsection*{{Explicit definition}}\label{ExplicitDefinition}

We unwind what the above amounts to more explicitly.

Let $(\mathbb{C},J)$ be an [[internal site]] in $\mathcal{S}$, i.e. an [[internal category]] $\mathbb{C}$ equipped with an internal [[coverage]] $J$. Let $\mathcal{S}^{\mathbb{C}^{op}}$ be the topos of [[internal diagram|internal diagrams]] on $\mathbb{C}^{op}$.

\begin{defn}
\label{}\hypertarget{}{}
\begin{enumerate}%
\item An \textbf{internal presheaf} on $\mathbb{C}$ is an internal diagram $F \in \mathcal{S}^{\mathbb{C}^{op}}$.


\item An \textbf{internal sheaf} on $\mathbb{C}$ (with respect to $J$) is an internal presheaf on $\mathbb{C}$ satisfying one of the following equivalent conditions:

\begin{enumerate}%
\item $F$ satisfies the \href{/nlab/show/sheaf#GeneralComponentwiseDefinition}{usual sheaf condition} interpreted in the [[internal language]] of $\mathcal{S}$.
\item $F$ is a $j$-sheaf for the [[Lawvere-Tierney topology]] on $\mathcal{S}^{\mathbb{C}^{op}}$ induced by $J$. (The equivalence is because the usual proof for $\mathcal{S} =$[[Set]] is constructive and can thus be internalized in an arbitrary topos.)

\end{enumerate}


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

Let $\mathcal{S}$ and $\mathbb{C}$ be as above.

\begin{prop}
\label{}\hypertarget{}{}
The category of internal presheaves $PSh(\mathbb{C}, \mathcal{S})$ is a [[topos]].

\end{prop}
This appears as (\hyperlink{Johnstone}{Johnstone, cor. B.2.3.17}).

\begin{prop}
\label{}\hypertarget{}{}
Let $f : \mathbb{C} \to \mathbb{D}$ be an [[internal functor]]. Write $\bar f : \bar \mathbb{C} \to \bar \mathbb{D}$ for the corresponding morphism in $\hat Sh_2(\mathcal{S}, can)$. Precomposition with this morphism induces a [[functor]] of internal presheaf catgeories

\begin{displaymath}
f^* \colon PSh(\mathbb{D}, \mathcal{S}) 
    \to 
   PSh(\mathbb{C}, \mathcal{S})
  \,.
\end{displaymath}
This is the [[inverse image]] of a [[geometric morphism]] of [[toposes]]

\begin{displaymath}
f \colon PSh(\mathbb{C}, \mathcal{S})
   \to 
  PSh(\mathbb{D}, \mathcal{S})
  \,.
\end{displaymath}
\end{prop}
This appears as (\hyperlink{Johnstone}{Johnstone, cor. B.2.3.22}).

The global section functors of internal sheaf toposes in $\mathcal{S}$ are [[bounded geometric morphisms]] over $\mathcal{S}$.

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

\begin{itemize}%
\item [[internal diagram]]

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

Internal presheaves in a [[Grothendieck topos]] are discussed in Section V.7 of

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

\end{itemize}
and in section B2.3 of

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

\end{itemize}
In these references internal presheaves are introduced in components as in the \hyperlink{ExplicitDefinition}{explicit definition} above. The equivalence to the \hyperlink{InTermsOf2Sheaves}{abstract formulation} above, in terms of morphisms between 2-sheaves, follows for instance with (\hyperlink{Johnstone}{Johnstone, lemma B.2.3.13}).

category: sheaf theory [[!redirects internal sheaves]] [[!redirects internal presheaf]] [[!redirects internal presheaves]] [[!redirects category of internal sheaves]] [[!redirects category of internal presheaves]] [[!redirects categories of internal sheaves]] [[!redirects categories of internal presheaves]] [[!redirects internal presheaf topos]] [[!redirects internal sheaf topos]]

[[!redirects internal presheaf toposes]] [[!redirects internal sheaf toposes]]



\end{document}