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\begin{document}

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\section*{concrete sheaf}

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

\hypertarget{discrete_and_concrete_objects}{}\paragraph*{{Discrete and concrete objects}}\label{discrete_and_concrete_objects}

[[!include discrete and concrete objects - 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{on_a_concrete_site}{On a concrete site}\dotfill \pageref*{on_a_concrete_site} \linebreak
\noindent\hyperlink{in_a_local_topos}{In a local topos}\dotfill \pageref*{in_a_local_topos} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{SliceOverConcreteObject}{Slice topos over a concrete object}\dotfill \pageref*{SliceOverConcreteObject} \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}

When regarding a [[sheaf]] as a [[space]] defined by how it is probed by test spaces, a \emph{concrete sheaf} is a generalized space that has (at least) an \emph{underlying [[set]] of points} out of which it is built.

So a concrete sheaf models a [[space]] that is given by a set of points and a choice of which morphisms of [[set]]s from concrete test spaces into it count as ``structure preserving'' (e.g. count as smooth, when the sheaf models a [[smooth space]]).

More in the intrinsic language of [[sheaves]], a \emph{concrete sheaf} is a [[sheaf]] on a [[concrete site]] $C$ that, while perhaps not [[representable functor|representable]], is ``quasi-representable'' in that it is a [[subobject]] of a sheaf of the form

\begin{displaymath}
U \mapsto Hom_{Set}(|U|, S)
\end{displaymath}
where $S$ is a [[set]] and $|U|$ is the [[set]] $|U| \coloneqq Hom_C({*}, U)$ of points underlying the object $U$ in the [[concrete site]] $C$.

For the category of concrete sheaves $Conc(Sh(C))$, the [[global sections]] functor $Hom_{Conc(Sh(C))}(1,-):Conc(Sh(C))\to Set$ is [[faithful functor|faithful]], i.e. $Conc(Sh(C))$ is a [[concrete category]].

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

We discuss two definitions: the first one is more elementary and describes concrete sheaves explicitly in terms of properties of the underlying [[site]].

The second one is more abstract and more general, and describes them entirely [[topos theory|topos theoretically]].

\hypertarget{on_a_concrete_site}{}\subsubsection*{{On a concrete site}}\label{on_a_concrete_site}

\begin{udef}
A \textbf{[[concrete site]]} is a [[site]] $C$ with a [[terminal object]] $*$ such that

\begin{enumerate}%
\item the functor $Hom_C(*,-) : C \to Set$ is a [[faithful functor]];


\item for every [[coverage|covering family]] $\{f_i : U_i \to U\}$ in $C$ the morphism

\begin{displaymath}
\coprod_i Hom_C(*,f_i) : \coprod_i Hom_C(*, U_i) \to Hom_C(*, U)
\end{displaymath}
is [[surjective]].



\end{enumerate}
\end{udef}
For $X \in PSh(C)$ any [[presheaf]], write

\begin{displaymath}
\tilde X_U : X(U) \to Hom_{Set}(Hom_C(*,U), X(*))
\end{displaymath}
for the [[adjunct]] of the restriction map

\begin{displaymath}
X(U) \times Hom_C(*,U) \to X(*)
  \,,
\end{displaymath}
which in turn is the adjunct of the component map of the functor

\begin{displaymath}
X_{*,U} . Hom_C(*,U) \to Hom_{Set}(X(U), X(*))
  \,.
\end{displaymath}
\hypertarget{definition_3}{}\paragraph*{{Definition}}\label{definition_3}

A [[presheaf]] $X : C^{op} \to Set$ on a [[concrete site]] is a \textbf{concrete presheaf} if for each $U \in C$ the map $\tilde X_U : X(U) \to Hom_{Set}(Hom_C(*,U), X(*))$ is [[injective]].

A \textbf{concrete sheaf} is a presheaf that is both concrete and a [[sheaf]].

So a concrete presheaf $X$ is a [[subobject]] of the presheaf $U \mapsto Hom_{Set}(Hom_C(*,U), X(*))$.

Write $Conc(Sh(C)) \hookrightarrow Sh(C)$ for the [[full subcategory]] of the [[category of sheaves]] on concrete sheaves.

\hypertarget{in_a_local_topos}{}\subsubsection*{{In a local topos}}\label{in_a_local_topos}

A more abstract perspective on the \hyperlink{ConcSheafOnConcSite}{previous definition} is obtained by noticing the following.

\begin{ulemma}
The [[category of sheaves]] on a [[concrete site]] is a [[local topos]].

\end{ulemma}
\begin{proof}
Taking $U=*$ in the second condition defining a concrete site implies that any [[covering]] family of $*$ contains a [[split epimorphism]], or equivalently that the only covering [[sieve]] of $*$ is the maximal sieve consisting of all morphisms with [[target]] $*$. This means that a concrete site is in particular a [[local site]], which implies that its topos of sheaves is a [[local topos]].

\end{proof}
In fact, we can formulate the definition of concrete sheaf inside any [[local topos]] $E$ over any base topos $S$:

\begin{udef}
Let

\begin{displaymath}
(Disc \dashv \Gamma \dashv Codisc) : E \stackrel{\stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}}}{\underset{Codisc}{\leftarrow}} S
\end{displaymath}
be a [[local geometric morphism]]. Since then by definition $S \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{Codisc}{\hookrightarrow}} E$ is a [[subtopos]] the morphisms $V = \Gamma^{-1}(isos(S)) \subset Mor E$ that are inverted by $\Gamma$ are the [[local isomorphism]]s with respect to which the objects of $S$ are [[sheaves]]/$V$-[[local object]]s in $E$.

The \textbf{concrete sheaves} are the objects of $E$ that are the $V$-[[separated objects]].

\end{udef}
\begin{uprop}
For $E = Sh(C) \stackrel{\Gamma = Hom(*,-)}{\to} Set$ the [[category of sheaves]] on a [[concrete site]], this is equivalent to the \hyperlink{ConcSheafOnConcSite}{previous definition}.

\end{uprop}
\begin{proof}
Since $C$ is concrete, in the [[global sections]] [[geometric morphism]] $(Disc,\Gamma)\colon Sh(C) \to Set$, the [[direct image]] $\Gamma$ is evaluation on the point: $X\mapsto X(*)$. The further [[right adjoint]] $Codisc \colon Set\to Sh(C)$, sends a set $A$ to the functor $U\mapsto Hom_{Set}(Hom_C(*,U),A)$. Moreover, this right adjoint $Codisc$ is [[full and faithful functor|fully faithful]] and thus embeds $Set$ as a [[subtopos]] of $Sh(C)$.

We observe that $(\Gamma \dashv Codisc) : Set \to  Sh(C)$ is the [[localization]] of $Sh(C)$ at the set $\{Disc \Gamma U \to U | U \in C\}$ of counits of the adjunction $(Disc \dashv \Gamma)$ on representables: because if for $X \in Sh(C)$ we have that

\begin{displaymath}
\begin{aligned}
     Hom_{Sh(C)}(Disc \Gamma U \to U, X) 
     & = (X(U) \to Hom_{Set}(\Gamma(U), \Gamma(X)))
     \\
     & = (X(U) \to Hom_{Set}(Hom_C(*,U)), X(*))
  \end{aligned}
\end{displaymath}
is an [[isomorphism]], then clearly $X = Codisc(X(*))$.

On the other hand, comparison with the \hyperlink{ConcSheafOnConcSite}{previous definition} shows that this is a [[monomorphism]] precisely if $X$ is a concrete sheaf. But this is also the definition of a separated object.

\end{proof}
So the concrete sheaves on $C$ are precisely the [[separated presheaf|separated objects]] for this [[Lawvere-Tierney topology]] on $Sh(C)$ that corresponds to the subtopos $Codisc : S \hookrightarrow Sh(C)$.

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Let $\Gamma : E \to S$ be a [[local topos]]. From the definition of concrete sheaves as [[separated presheaves]] it follows immediately that

\begin{uprop}
The category of concrete sheaves $Conc_\Gamma(E)$ forms a [[reflective subcategory]] of $E$

\begin{displaymath}
S 
   \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{Codisc}{\hookrightarrow}}
  Conc_\Gamma(E) 
    \stackrel{\overset{Conc}{\leftarrow}}{\hookrightarrow}
  E
\end{displaymath}
which is a [[quasitopos]].

The left adjoint $Conc$ is \textbf{concretization} which sends a sheaf $X$ to the [[image]] sheaf

\begin{displaymath}
Conc X : U \mapsto Im(X(U) \to Hom_{Set}(Hom_C(*,U), X(*)))
  \,.
\end{displaymath}
\end{uprop}
\hypertarget{SliceOverConcreteObject}{}\subsubsection*{{Slice topos over a concrete object}}\label{SliceOverConcreteObject}

Let $(Disc \dashv \Gamma \dashv coDisc) :  \mathcal{E} \to \mathcal{S}$ be a [[Grothendieck topos]] that is a [[local topos]] over $\mathcal{S}$ and let $X \in \mathcal{E}$ be a concrete object, equivalently an object such that the $(\Gamma \dashv coDisc)$-[[unit of an adjunction|counit]] $X \to coDisc \Gamma U$ is a [[monomorphism]].

We discuss properties of the [[over-topos]] $\mathcal{E}/X$.

Notice that

\begin{displaymath}
e_0 \coloneqq (\Gamma \dashv coDisc) : \mathcal{S} \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{coDisc}{\to}} \mathcal{E}
\end{displaymath}
is the canonical [[point of a topos|topos point]] of $\mathcal{E}$.

\begin{uprop}
For every [[global element]] $(x \in \Gamma(X)) : * \to X$ (for every $X \in \mathcal{E}$) there is a [[point of a topos|topos point]] of the form

\begin{displaymath}
(e_0,x)
  : 
  \mathcal{S} 
    \stackrel{\overset{x^*}{\leftarrow}}{\underset{x_*}{\to}}
  \mathcal{S}/\Gamma(X)
    \stackrel{\overset{\Gamma/X}{\leftarrow}}{\underset{coDisc/X}{\to}}
  \mathcal{E}/X
  \,.
\end{displaymath}
\end{uprop}
This is discussed in detail at .

\begin{uprop}
\textbf{(relative concretization)}

Let $X \in \mathcal{E}$ be concrete. Then the image under the $coDisc/X \circ \Gamma/X$-[[monad]] of any object $(A \to X) \in \mathcal{E}/X$ is an object $(\tilde A \to X)$ with $\tilde A$ being concrete.

This $\tilde A$ is the finest concrete sheaf structure on $\Gamma A$ that extends $\Gamma A \to \Gamma X$ to a morphism of concrete sheaves.

\end{uprop}
\begin{proof}
By definition of the  we have that $coDisc/X \circ \Gamma/X (A \stackrel{f}{\to} X)$ is the [[pullback]] $\tilde A \to X$ in

\begin{displaymath}
\itexarray{
    \tilde A &\to & coDisc \Gamma A
    \\
    \downarrow && \downarrow^{\mathrlap{coDisc \Gamma A}}
    \\
    X &\to& coDisc \Gamma X
  }
  \,,
\end{displaymath}
where the bottom morphism is the $(\Gamma \dashv coDisc)$-[[unit of an adjunction|unit]]. Since this is a [[monomorphism]] by assumption on $X$ it follows that $\tilde A\to coDisc \Gamma A$ is a monomorphism. Since $coDisc$ is a [[full and faithful functor]] by assumption on $\mathcal{E}$ and $\Gamma$ is a [[right adjoint]] it follows that the [[adjunct]] $\Gamma \tilde A \to \Gamma coDisc\Gamma A \stackrel{\simeq}{\to} \Gamma A$ is a monomorphism, as is its image $coDisc \Gamma \tilde A\to coDisc \Gamma A$ under the [[right adjoint]] $coDisc$.

Then by the  we have a commuting diagram

\begin{displaymath}
\itexarray{
    && coDisc \Gamma \tilde A
    \\
    & \nearrow & \downarrow
    \\
    \tilde A &\to& coDisc \Gamma A 
  }
  \,,
\end{displaymath}
where the bottom and the right morphisms are [[monomorphism]]s. Therefore also the diagonal morphism, the $(\Gamma \dashv coDisc)$-unit on $\tilde A$, is a monomorphism, and hence $\tilde A$ is concrete.

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

\begin{itemize}%
\item The concrete sheaves on the [[concrete site]] [[CartSp]] are the [[diffeological space]]s.

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

\begin{itemize}%
\item \textbf{concrete sheaf}


\item [[concrete (∞,1)-sheaf]]



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

The notion of quasitoposes of concrete sheaves goes back to

\begin{itemize}%
\item [[Eduardo Dubuc]], \emph{Concrete quasitopoi} , Lecture Notes in Math. 753 (1979), 239--254

\end{itemize}
and is further developed in

\begin{itemize}%
\item [[Eduardo Dubuc]], L. Espanol, \emph{Quasitopoi over a base category} (\href{http://arxiv.org/abs/math.CT/0612727}{arXiv:math.CT/0612727}) .

\end{itemize}
A review of categories of concrete sheaves, with special attention to sheaves on [[CartSp]], i.e. to [[diffeological space]]s is in

\begin{itemize}%
\item [[John Baez]], [[Alex Hoffnung]], \emph{Convenient Categories of Smooth Spaces}, Trans. Amer. Math. Soc. 363 (11), 2011 (\href{http://arxiv.org/abs/0807.1704}{arXiv})

\end{itemize}
The characterization of concrete sheaves in terms of the extra right adjoint of a [[local topos]] originated in \href{http://nforum.ncatlab.org/comments.php?DiscussionID=1955}{discussion} with [[David Carchedi]].

[[!redirects concrete sheaves]] [[!redirects concrete presheaf]] [[!redirects concrete presheaves]]



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