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\begin{document}

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\section*{closed monoidal structure on presheaves}

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

\hypertarget{monoidal_categories}{}\paragraph*{{Monoidal categories}}\label{monoidal_categories}

[[!include monoidal categories - contents]]

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category 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{definition_in_terms_of_homs_of_direct_images}{Definition in terms of homs of direct images}\dotfill \pageref*{definition_in_terms_of_homs_of_direct_images} \linebreak
\noindent\hyperlink{relation_of_the_two_definitions}{Relation of the two definitions}\dotfill \pageref*{relation_of_the_two_definitions} \linebreak
\noindent\hyperlink{presheaves_over_a_monoidal_category}{Presheaves over a monoidal category}\dotfill \pageref*{presheaves_over_a_monoidal_category} \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}

As every [[topos]], a [[category of presheaves]] is [[cartesian monoidal category|cartesian]] [[closed monoidal category|closed monoidal]].

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

\begin{prop}
\label{CartesianClosedMonoidalnessOfCategoriesOfPresheaves}\hypertarget{CartesianClosedMonoidalnessOfCategoriesOfPresheaves}{}
\textbf{([[cartesian closed category|cartesian closure]] of [[categories of presheaves]])}

Let $\mathcal{C}$ be a [[category]] and write $[\mathcal{C}^{op}, Set]$ for its [[category of presheaves]].

This is

\begin{enumerate}%
\item a [[cartesian monoidal category]], whose [[Cartesian product]] is given objectwise in $\mathcal{C}$ by the [[Cartesian product]] in [[Set]]:

for $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$, their [[Cartesian product]] $\mathbf{X} \times \mathbf{Y}$ exists and is given by

\begin{displaymath}
\mathbf{X} \times \mathbf{Y}
  \;\;\colon\;\;\phantom{A}
  \itexarray{
    c_1 &\mapsto& \mathbf{X}(c_1) \times \mathbf{Y}(c_1)
    \\
    {}^{\mathllap{f}}\big\downarrow 
      &&
    \big\uparrow^{ \mathrlap{ \mathbf{X}(f) \times \mathbf{Y}(f) } }
    \\
    c_2 &\mapsto& \mathbf{X}(c_2) \times \mathbf{Y}(c_2)
  }
\end{displaymath}

\item a [[cartesian closed category]], whose [[internal hom]] is given for $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$ by

\begin{displaymath}
[\mathbf{X}, \mathbf{Y}]
  \;\;\colon\;\;\phantom{A}
  \itexarray{
    c_1 &\mapsto& Hom_{[\mathcal{C}^{op}, Set]}( y(c_1) \times \mathbf{X}, \mathbf{y} )
    \\
    {}^{ \mathllap{ f } }\big\downarrow 
      &&
    \big\uparrow^{ \mathrlap{ Hom_{[\mathcal{C}^{op}, Set]}( y(f) \times \mathbf{X}, \mathbf{y} ) } }
    \\
    c_2 &\mapsto& Hom_{[\mathcal{C}^{op}, Set]}( y(c_2) \times \mathbf{X}, \mathbf{y} )       
  }
\end{displaymath}
Here $y \;\colon\; \mathcal{C} \to [\mathcal{C}^{op}, Set]$ denotes the [[Yoneda embedding]] and $Hom_{[\mathcal{C}^{op}, Set]}(-,-)$ is the [[hom-functor]] on the [[category of presheaves]].



\end{enumerate}
\end{prop}
(e.g. \hyperlink{MacLaneMoerdijk}{MacLane-Moerdijk, pages 46-47}).

\begin{proof}
The first statement is a special case of the general fact that [[limits of presheaves are computed objectwise]].

For the second statement, first assume that $[\mathbf{X}, \mathbf{Y}]$ does exist. Then by the [[adjoint functor|hom-adjunction isomorphism]] we have for any other presheaf $\mathbf{Z}$ a [[natural isomorphism]] of the form

\begin{equation}
Hom_{[\mathcal{C}^{op}, Set]}(\mathbf{Z}, [\mathbf{X},\mathbf{Y}]) 
    \;\simeq\; 
  Hom_{[\mathcal{C}^{op}, Set]}(\mathbf{Z} \times \mathbf{X}, \mathbf{Y})
  \,.
\label{InternalHomIsoInPresheaves}\end{equation}
This holds in particular for $\mathbf{Z} = y(c)$ a [[representable functor]] (i.e. in the [[essential image]] of the [[Yoneda embedding]]) and so the [[Yoneda lemma]] implies that if it exists, then $[\mathbf{X}, \mathbf{Y}]$ must have the claimed form:

\begin{displaymath}
\begin{aligned}
    [\mathbf{X}, \mathbf{Y}](c)
    & \simeq
    Hom_{[\mathcal{C}^{op}, Set]}( y(c), [\mathbf{X}, \mathbf{Y}] )
    \\
    & \simeq Hom_{ [\mathcal{C}^{op}, Set] }( y(c) \times \mathbf{X}, \mathbf{Y} )
    \,.
  \end{aligned}
\end{displaymath}
Hence it remains to show that this formula does make \eqref{InternalHomIsoInPresheaves} hold generally.

For this we use the equivalent definition of [[adjoint functors]] in terms of the [[adjunction counit]] providing a system of [[universal arrows]].

Define a would-be [[adjunction counit]], which here is called an \emph{[[evaluation]]} morphism, by

\begin{displaymath}
\itexarray{
    \mathbf{X} \times [\mathbf{X} , \mathbf{Y}]
    &\overset{ev}{\longrightarrow}&
    \mathbf{Y}
    \\
    \mathbf{X}(c) \times Hom_{[\mathcal{C}^{op}, Set]}(y(c) \times \mathbf{X}, \mathbf{Y})
    &\overset{ev_c}{\longrightarrow}&
    \mathbf{Y}(c)
    \\
    (x, \phi) &\mapsto& \phi_c( id_c, x )
  }
\end{displaymath}
Then it remains to show that for every morphism of presheaves of the form $\mathbf{X} \times \mathbf{A} \overset{\phantom{A}f\phantom{A}}{\longrightarrow} \mathbf{Y}$ there is a \emph{unique} morphism $\widetilde f \;\colon\; \mathbf{A} \longrightarrow [\mathbf{X}, \mathbf{Y}]$ such that

\begin{equation}
\itexarray{
    \mathbf{X} \times \mathbf{A}
    && \overset{ \mathbf{X} \times \widetilde f  }{\longrightarrow} && 
    \mathbf{X} \times [\mathbf{X}, \mathbf{Y}]
    \\
    & {}_{\mathllap{ \mathrlap{f} }}\searrow 
      && 
    \swarrow_{ \mathrlap{  ev } }
    \\
    && \mathbf{Y}
  }
\label{UniversalArrowConditionForEvaluationMapInPresheaves}\end{equation}
The [[commuting diagram|commutativity]] of this diagram means in components at $c \in \mathcal{C}$ that, that for all $x \in \mathbf{X}(c)$ and $a \in \mathbf{A}(c)$ we have

\begin{displaymath}
\begin{aligned}
    ev_c( x, \widetilde f_c(a) )
    & \coloneqq
    (\widetilde f_c(a))_c( id_c, x )
    \\
    & = f_c( x, a )
  \end{aligned}
\end{displaymath}
Hence this fixes the component $\widetilde f_c(a)_c$ when its first argument is the [[identity morphism]] $id_c$. But let $g \;\colon\; d \to c$ be any morphism and chase $(id_c, x )$ through the naturality diagram for $\widetilde f_c(a)$:

\begin{displaymath}
\itexarray{
    Hom_{\mathcal{C}}(c,c) \times \mathbf{X}(c)
    &\overset{ (\widetilde f_c(a))_c }{\longrightarrow}&
    \mathbf{Y}(c) 
    \\
    {}^{\mathllap{ g^\ast }}\big\downarrow
     &&
    \big\downarrow^{\mathrlap{ g^\ast }}
    \\
    Hom_{\mathcal{C}}(d,c) \times \mathbf{X}(d)
    &\overset{ (\widetilde f_c(a))_d }{\longrightarrow}&
    \mathbf{Y}(d) 
  }
  \phantom{AAAA}
  \itexarray{
    \{ (id_c, x ) \} &\longrightarrow& \{ f_c( x, a ) \}
    \\
    \big\downarrow && \big\downarrow 
    \\
    \{ (g, g^\ast(x)) \}  &\longrightarrow& \{ f_d( g^\ast(x), g^\ast(a) ) \}
  }
\end{displaymath}
This shows that $(\widetilde f_c(a))_d$ is fixed to be given by

\begin{equation}
(\widetilde f_c(a))_d( g, x' )
  \;=\;
  f_d( x', g^\ast(a) )
\label{ComponentFormulaForEvaluationMapInPresheaves}\end{equation}
at least on those pairs $(g,x')$ such that $x'$ is in the image of $g^\ast$.

But, finally, $(\widetilde f_c(a))_d$ is also natural in $c$

\begin{displaymath}
\itexarray{
    \mathbf{A}(c)
    &\overset{ \widetilde f_c }{\longrightarrow}& [\mathbf{X},\mathbf{Y}](c)
    \\
    {}^{\mathllap{g^\ast}}\big\downarrow && \big\downarrow^{\mathrlap{g^\ast}}
    \\
    \mathbf{A}(d)
    &\overset{ \widetilde f_d }{\longrightarrow}& [\mathbf{X},\mathbf{Y}](d)
  }
\end{displaymath}
which implies that \eqref{ComponentFormulaForEvaluationMapInPresheaves} must hold generally. Hence naturality implies that \eqref{UniversalArrowConditionForEvaluationMapInPresheaves} indeed has a unique solution.

\end{proof}
\hypertarget{definition_in_terms_of_homs_of_direct_images}{}\subsection*{{Definition in terms of homs of direct images}}\label{definition_in_terms_of_homs_of_direct_images}

Often another, equivalent, expression is used to express the internal hom of presheaves:

Let $X$ be a [[site|pre-site]] with underlying [[category]] $S_X$. Recall from the discussion at [[site]] that just means that we have a category $S_X$ on which we consider [[presheaf|presheaves]] $F \in PSh(S_X) := [S_X^{op}, Set]$, but that it suggests that

\begin{itemize}%
\item to each object $U \in PSh(X)$ and in particular to each $U \in S_X \hookrightarrow PSh(X)$ there is naturally associated the pre-site $U$ with underlying category the [[comma category]] $S_U = (Y/Y(U))$;


\item that the canonical [[stuff, structure, property|forgetful]] [[functor]] $j^t_{U \to X} :  S_U \to S_X$, which can be thought of as a [[site|morphism of pre-sites]] $j_{U \to X} : X \to U$ induces the [[direct image]] functor $(j_{U \to X})_* : PSh(X) \to PSh(U)$ which we write $F \mapsto F|_U$.



\end{itemize}
Then in these terms the above \textbf{internal hom} for presheaves

\begin{displaymath}
hom : PSh(X)^{op} \times PSh(X) \to PSh(X)
\end{displaymath}
is expressed for all $F,G \in PSh(X)$ by

\begin{displaymath}
hom(F,G) = U \mapsto Hom_{PSh(U)}(F|_U, G|_U)
  \,.
\end{displaymath}
\hypertarget{relation_of_the_two_definitions}{}\subsection*{{Relation of the two definitions}}\label{relation_of_the_two_definitions}

To see the equivalence of the two definitions, notice

\begin{itemize}%
\item that by the [[Yoneda lemma]] we have that $S_U$ is simply the [[over category]] $S_U = S_X/U$;
\item by the general properties of [[functors and comma categories]] there is an equivalence $PSh(S_X/U) \simeq PSh(S_X)/y(U)$;
\item which identifies the functor $(-)|_U : PSh(S_X) \to PSh(S_U)$ with the functor $((-)\times y(U) \stackrel{p_2}{\to} y(U)) : PSh(S_X) \to PSh(S_X)/y(U)$;
\item and that $Hom_{PSh(S_X)/y(U)}(y(U) \times F, y(U) \times G) \simeq Hom_{PSh(S_X)}(y(U) \times F, G)$.

\end{itemize}
\hypertarget{presheaves_over_a_monoidal_category}{}\subsection*{{Presheaves over a monoidal category}}\label{presheaves_over_a_monoidal_category}

It is worth noting that in the case where $X$ is itself a [[monoidal category]] $(X, \otimes, I)$, $Psh(X)$ is equipped with another (bi)closed monoidal structure given by the [[Day convolution]] product and its componentwise right adjoints. Let $F$ and $G$ be two presheaves over $X$. Their tensor product $F \star G$ can be defined by the following [[coend]] formula:

\begin{displaymath}
F\star G = U \mapsto \int^{U_1,U_2\in X} Hom_X(U, U_1\otimes U_2) \times F(U_1) \times G(U_2)
\end{displaymath}
Then we can define two right adjoints

\begin{displaymath}
F\star - \dashv F \backslash -
\qquad -\star G \dashv - / G
\end{displaymath}
by the following [[end]] formulas:

\begin{displaymath}
F \backslash H = V \mapsto \int_{U\in X} F(U) \to H(U\otimes V)
\end{displaymath}
\begin{displaymath}
H / G = U \mapsto \int_{V\in X} G(V) \to H(U\otimes V)
\end{displaymath}
In the case where the monoidal structure on $X$ is cartesian, the induced closed monoidal structure on $Psh(X)$ coincides with the cartesian closed structure described in the previous sections.

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

\begin{itemize}%
\item The cartesian closure restricts from presheaves to [[categories of sheaves]] (e.g. \hyperlink{MacLaneMoerdijk}{MacLane-Moerdijk, p. 136-138})

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

The first definition is discussed for instance in section I.6 of

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

\end{itemize}
The second definition is discussed for instance in section 17.1 of

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

\end{itemize}
[[!redirects ccc structure on presheaves]] [[!redirects cartesian closed structure on presheaves]] [[!redirects cartesian closed monoidal structure on presheaves]]

[[!redirects cartesian closure of categories of presheaves]] [[!redirects categories of presheaves are cartesian closed]]



\end{document}