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\begin{document}

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\section*{internal hom}

\begin{quote}%
This page discusses the general concept of mapping spaces and internal homs. For mapping spaces in topology, see at \emph{[[compact-open topology]]}.


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

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

[[!include category theory - contents]]

\hypertarget{mapping_space}{}\paragraph*{{Mapping space}}\label{mapping_space}

[[!include mapping space - 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{EvaluationMap}{Evaluation map}\dotfill \pageref*{EvaluationMap} \linebreak
\noindent\hyperlink{CompositionMap}{Composition map}\dotfill \pageref*{CompositionMap} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{BasicProperties}{Basic properties}\dotfill \pageref*{BasicProperties} \linebreak
\noindent\hyperlink{relation_to_function_types}{Relation to function types}\dotfill \pageref*{relation_to_function_types} \linebreak
\noindent\hyperlink{induced_monad_state_monad}{Induced monad (state monad)}\dotfill \pageref*{induced_monad_state_monad} \linebreak
\noindent\hyperlink{StableSplitting}{Stable splitting}\dotfill \pageref*{StableSplitting} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{in_sets}{In sets}\dotfill \pageref*{in_sets} \linebreak
\noindent\hyperlink{in_simplicial_sets}{In simplicial sets}\dotfill \pageref*{in_simplicial_sets} \linebreak
\noindent\hyperlink{InASheafTopos}{In a sheaf topos or $(\infty,1)$-sheaf $(\infty,1)$-topos}\dotfill \pageref*{InASheafTopos} \linebreak
\noindent\hyperlink{ExampleInSliceCategories}{In slice categories}\dotfill \pageref*{ExampleInSliceCategories} \linebreak
\noindent\hyperlink{for_smooth_spaces_and_smooth_groupoids}{For smooth spaces and smooth $\infty$-groupoids}\dotfill \pageref*{for_smooth_spaces_and_smooth_groupoids} \linebreak
\noindent\hyperlink{for_chain_complexes}{For chain complexes}\dotfill \pageref*{for_chain_complexes} \linebreak
\noindent\hyperlink{for_super_vector_spaces}{For super vector spaces}\dotfill \pageref*{for_super_vector_spaces} \linebreak
\noindent\hyperlink{for_banach_spaces}{For Banach spaces}\dotfill \pageref*{for_banach_spaces} \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}

For $\mathcal{C}$ a [[category]] and $X, Y \in \mathcal{C}$ two [[objects]], the \emph{internal hom} $[X,Y] \in \mathcal{C}$ from $X$ to $Y$ is, if it exists, another [[object]] of $\mathcal{C}$ which behaves like the ``object of [[morphisms]]'' from $X$ to $Y$. In other words it is, if it exists, an [[internalization|internal version]] of the ordinary [[hom set]] $\mathcal{C} \in Set$ or more generally [[hom object]] $\mathcal{C}(X, Y) \in \mathcal{V}$ of a [[locally small category]] or $\mathcal{V}$-[[enriched category]].

One way to make this precise starts by mimicking a property of the [[function set]] $[X,Y] = \{f : X \to Y\}$ of [[functions]] between two [[sets]] $X$ and $Y$: this set is characterized by the fact that for any other set $S$, the functions $S \to [X,Y]$ are in [[natural bijection]] with the functions $S \times X \to Y$ out of the [[cartesian product]] of $S$ with $X$. That is: for each set $X$, the [[functor]] $(-) \times X$ has a [[right adjoint]], given by the construction $[X,-]$.

One can verbalize this thus: \emph{taking the cartesian product with the set $X$} is left-adjoint to \emph{taking the set of all functions out of $X$}.

This, then, is, generally, the definition of \emph{internal hom} in any [[cartesian monoidal category]] or in fact in any [[monoidal category]] $(\mathcal{C}, \otimes)$: the [[right adjoint]] $[X,-]$ to the given [[tensor product]] functor $(-)\otimes X$ for all objects $X$. It may or may not exist. If it exists, one says that $(\mathcal{C}, \otimes)$ is a \emph{[[closed monoidal category|closed]]} [[monoidal category]]. Explicity, the condition is that there is an [[isomorphism]]([[bijection]])

\begin{displaymath}
\mathcal{C}(A, [X,Z]) \stackrel{\simeq}{\to} \mathcal{C}(A \otimes X, Z)
\end{displaymath}
which is [[natural isomorphism|natural]] in all three [[variables]]. (The rightward map here is often called \textbf{[[currying]]}, especially in a [[closed monoidal category]] (and more especially for the $\lambda$-[[lambda-calculus|calculus]]).)

In particular this implies that in a closed monoidal category the external hom is re-obtained from the internal hom as its set of [[generalized elements]] out of the [[unit object|tensor unit]] $I \in \mathcal{C}$ in that

\begin{displaymath}
\frac{I \to [X,Y]}{X \to Y}
\end{displaymath}
using that $I \otimes X \simeq X$ by definition of the tensor unit.

Here ``closed'' in ``[[closed monoidal category]]'' is in the sense that forming ``hom-sets'' does not lead ``out of the category''. In fact the internal hom of a [[cartesian monoidal category]] is indeed the hom as seen in the \emph{[[internal logic]]} of that category (the \emph{[[function type]]}).

More generally, one can consider objects that satisfy some basic [[universal properties]] that an internal hom should satisfy even in the absence of a [[monoidal category|monoidal structure]]. If such objects exist one speaks therefore just of a \emph{[[closed category]]}. Every [[closed category]] may be seen as a category [[enriched category|enriched]] over itself. Accordingly, an internal hom is after all a special case of a [[hom-object]], for the special case of this enrichment over itself.

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

\begin{defn}
\label{ClosedMonoidalCategory}\hypertarget{ClosedMonoidalCategory}{}
\textbf{(internal hom)}

Let $(\mathcal{C}, \otimes)$ be a [[symmetric monoidal category|symmetric]] [[monoidal category]]. An \textbf{internal hom} in $\mathcal{C}$ is a [[functor]]

\begin{displaymath}
[-,-] : \mathcal{C}^{op} \times \mathcal{C} \to \mathcal{C}
\end{displaymath}
such that for every [[object]] $X \in \mathcal{C}$ we have a pair of [[adjoint functors]]

\begin{displaymath}
((-) \otimes X \dashv [X, -]) : \mathcal{C} \to \mathcal{C}
  \,.
\end{displaymath}
If this exists, $(\mathcal{C}, \otimes)$ is called a \emph{[[closed monoidal category]]}.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
If the monoidal category $\mathcal{C}$ in Def. \ref{ClosedMonoidalCategory} is not [[symmetric monoidal category|symmetric]], there is instead a concept of left- and right-internal hom.

\end{remark}
\hypertarget{EvaluationMap}{}\subsubsection*{{Evaluation map}}\label{EvaluationMap}

Let $(\mathcal{C}, \otimes)$ be a [[symmetric monoidal category|symmetric]] [[closed monoidal category]] (Def. \ref{ClosedMonoidalCategory}).

\begin{defn}
\label{EvalMap}\hypertarget{EvalMap}{}
For $X,Y \in \mathcal{C}$ two [[objects]], the \textbf{[[evaluation map]]}

\begin{displaymath}
eval_{X,Y} : [X,Y] \otimes X \to Y
\end{displaymath}
is the $((-)\otimes X \dashv [X,-])$-[[adjunct]] of the [[identity]] $id_{[X,Y]} : [X,Y] \to [X,Y]$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
If $\mathcal{C}$ is specifically a [[locally cartesian closed category]], then in terms of the [[type theory]] [[internal language]] of $\mathcal{C}$ the [[evaluation map]] is the [[categorical semantics]] of the [[dependent type]] which in [[type theory]] [[syntax]] is

\begin{displaymath}
f \colon X \to Y,\; x \colon X 
    \;\vdash\; 
   f(x) \colon Y
  \,,
\end{displaymath}
with \emph{[[function application]]} on the right.

\end{remark}
\hypertarget{CompositionMap}{}\subsubsection*{{Composition map}}\label{CompositionMap}

Let $(\mathcal{C}, \times)$ be a [[symmetric monoidal category|symmetric]] [[closed monoidal category]] (Def. \ref{ClosedMonoidalCategory})

\begin{defn}
\label{CompositionMorphism}\hypertarget{CompositionMorphism}{}
For $X, Y, Z \in \mathcal{C}$ three [[objects]], the \textbf{[[composition]] morphism}

\begin{displaymath}
\circ_{X,Y,Z} : 
  [Y, Z]
  \times
  [X, Y]
  \to 
  [X, Z]
\end{displaymath}
is the $((-)\times X \dashv [X,-])$-[[adjunct]] of the following composite of two [[evaluation maps]], def. \ref{EvalMap}:

\begin{displaymath}
[Y, Z] \times [X , Y] \times X 
    \stackrel{(id_{[Y,Z]}, eval_{X,Y})}{\to}
  [Y,Z] \times Y
    \stackrel{eval_{Y,Z}}{\to}
  Z
  \,.
\end{displaymath}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{BasicProperties}{}\subsubsection*{{Basic properties}}\label{BasicProperties}

The internal homs in a [[symmetric monoidal category|symmetric]] [[closed monoidal category]] (Def. \ref{ClosedMonoidalCategory}) happen to share all the key abstract properties of ordinary (``external'') [[hom-functors]], even though this is not completely manifest from Def. \ref{ClosedMonoidalCategory}:

\begin{prop}
\label{InternalHomBifunctor}\hypertarget{InternalHomBifunctor}{}
\textbf{([[internal hom]] [[bifunctor]])}

Let $\mathcal{C}$ be a [[symmetric monoidal category]] such that for each object $X \in \mathcal{C}$ the functor $X \otimes (-)$ has a [[right adjoint]] $[X,-]$. Then this is already equivalent to Def. \ref{ClosedMonoidalCategory}, in that there is a unique functor out of the [[product category]]of $\mathcal{C}$ with its [[opposite category]]

\begin{displaymath}
[-,-]
  \;\colon\;
  \mathcal{C}^{op} \times \mathcal{C}
    \longrightarrow
  \mathcal{C}
\end{displaymath}
such that for each $X \in \mathcal{C}$ it coincides with the [[internal hom]] $[X,-]$ as a functor in the second variable, and such that there is a [[natural isomorphism]]

\begin{displaymath}
Hom(X, [Y,Z])
  \;\simeq\;
  Hom(X \otimes Y, Z)
\end{displaymath}
which is natural not only in $X$ and $Z$, but also in $Y$.

\end{prop}
\begin{proof}
We have a natural isomorphism for each fixed $Y$, and hence in particular for fixed $Y$ and fixed $Z$. With this the statement follows directly by \href{https://ncatlab.org/nlab/show/adjoint%20functor#AdjointFunctorFromObjectwiseRepresentingObject}{this prop.} at \emph{[[adjoint functor]]}.

\end{proof}
In fact the 3-variable adjunction from Prop. \ref{InternalHomBifunctor} even holds internally:

\begin{prop}
\label{TensorHomAdjunctionIsoInternally}\hypertarget{TensorHomAdjunctionIsoInternally}{}
\textbf{(internal tensor/hom-adjunction)}

In a [[symmetric monoidal category|symmetric]] [[closed monoidal category]] (def. \ref{ClosedMonoidalCategory}) there are [[natural isomorphisms]]

\begin{displaymath}
[X \otimes Y, Z]
   \;\simeq\;
  [X, [Y,Z]]
\end{displaymath}
whose image under $Hom_{\mathcal{C}}(1,-)$ are the defining [[natural bijections]] of Prop. \ref{InternalHomBifunctor}.

\end{prop}
\begin{proof}
Let $A \in \mathcal{C}$ be any object. By applying the natural bijections from Prop. \ref{InternalHomBifunctor}, there are composite [[natural bijections]]

\begin{displaymath}
\begin{aligned}
    Hom_{\mathcal{C}}(A , [X \otimes Y, Z])
    & \simeq
    Hom_{\mathcal{C}}(A \otimes (X \otimes Y), Z)
    \\
    & \simeq
    Hom_{\mathcal{C}}((A \otimes X)\otimes Y, Z)
    \\
    & \simeq
    Hom_{\mathcal{C}}(A \otimes X, [Y,Z])
    \\
    & \simeq
    Hom_{\mathcal{C}}(A, [X, [Y,Z]])
  \end{aligned}
\end{displaymath}
Since this holds for all $A$, the [[fully faithful functor|fully faithfulness]] of the [[Yoneda embedding]] says that there is an isomorphism $[ X\otimes Y, Z ] \simeq [X, [Y,Z]]$. Moreover, by taking $A = 1$ in the above and using the left [[unitor]] isomorphisms $A \otimes (X \otimes Y) \simeq X \otimes Y$ and $A\otimes X \simeq X$ we get a [[commuting diagram]]

\begin{displaymath}
\itexarray{
    Hom_{\mathcal{C}}(1, [X\otimes Y, Z ))
      &\overset{\simeq}{\longrightarrow}&
    Hom_{\mathcal{C}}(1, [X, [Y,Z]])
    \\
    {}^{\mathllap{\simeq}}\downarrow
      &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    Hom_{\mathcal{C}}(X \otimes Y, Z)
     &\overset{\simeq}{\longrightarrow}&
    Hom_{\mathcal{C}}(X, [Y,Z])
  }
  \,.
\end{displaymath}
\end{proof}
Also the key respect of [[hom-functors]] for [[limits]] is inherited by [[internal hom]]-functors:

\begin{prop}
\label{InternalHomPreservesLimits}\hypertarget{InternalHomPreservesLimits}{}
\textbf{([[internal hom-functor preserves limits]])}

Let $\mathcal{C}$ be a [[symmetric monoidal category|symmetric]] [[closed monoidal category]] with [[internal hom]]-[[bifunctor]] $[-,-]$ (Prop. \ref{InternalHomBifunctor}). Then this bifunctor preserves [[limits]] in the second variable, and sends colimits in the first variable to limits:

\begin{displaymath}
[X, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Y(j)]
  \;\simeq\;
  \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [X, Y(j)]
\end{displaymath}
and

\begin{displaymath}
[\underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j),X]
  \;\simeq\;
  \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j),X]
\end{displaymath}
\end{prop}
\begin{proof}
For $X \in \mathcal{C}$ any object, $[X,-]$ is a [[right adjoint]] by definition, and hence preserves limits by \emph{[[adjoints preserve (co-)limits]]}.

For the other case, let $Y \;\colon\; \mathcal{L} \to \mathcal{C}$ be a [[diagram]] in $\mathcal{C}$, and let $C \in \mathcal{C}$ be any object. Then there are isomorphisms

\begin{displaymath}
\begin{aligned}
    Hom_{\mathcal{C}}(C, [ \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X ] )
    & \simeq
    Hom_{\mathcal{C}}( C \otimes \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X )
    \\
    & \simeq
    Hom_{\mathcal{C}}( \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} (C \otimes Y(j)), X )   
    \\
    & \simeq
    \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim}
    Hom_{\mathcal{C}}( (C \otimes Y(j)), X )    
    \\
    & \simeq
    \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim}
    Hom_{\mathcal{C}}( C, [Y(j), X] )    
    \\
    & \simeq
    Hom_{\mathcal{C}}( C, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j), X] )    
  \end{aligned}
\end{displaymath}
which are [[natural isomorphism|natural]] in $C \in \mathcal{C}$, where we used that the ordinary [[hom-functor]] respects (co)limits as shown (see at \emph{[[hom-functor preserves limits]]}), and that the [[left adjoint]] $C \otimes (-)$ preserves colimits (see at \emph{[[adjoints preserve (co-)limits]]}).

Hence by the [[fully faithful functor|fully faithfulness]] of the [[Yoneda embedding]], there is an isomorphism

\begin{displaymath}
\left[ \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X \right]  
  \overset{\simeq}{\longrightarrow}
  \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j), X] 
  \,.
\end{displaymath}
\end{proof}
\hypertarget{relation_to_function_types}{}\subsubsection*{{Relation to function types}}\label{relation_to_function_types}

The internal hom is the [[categorical semantics]] of what in [[type theory]] are [[function types]]

[[!include function type natural deduction - table]]

\hypertarget{induced_monad_state_monad}{}\subsubsection*{{Induced monad (state monad)}}\label{induced_monad_state_monad}

For each object $S$ the (internal hom $\dashv$ [[tensor product]])-[[adjunction]] induces a [[monad]] $[S, S \otimes (-)]$. In [[computer science]] this [[monad (in computer science)]] is called the \emph{[[state monad]]}.

\hypertarget{StableSplitting}{}\subsubsection*{{Stable splitting}}\label{StableSplitting}

In [[topology]] the [[stabilization]]/[[suspension spectrum]] $\Sigma^\infty  Maps(X,A)$ of [[mapping spaces]] $Maps(X,A)$ between suitable [[CW-complexes]] $X, A$ happens to decompose as a [[direct sum]] of [[spectra]] in a useful way, related to the expression of the [[Goodwillie derivatives]] of the functor $Maps(X,-)$.

For more on this see at \emph{[[stable splitting of mapping spaces]]}.

\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\hypertarget{in_sets}{}\subsubsection*{{In sets}}\label{in_sets}

In the category [[Set]] of [[sets]], regarded as a [[cartesian monoidal category]], the internal hom is given by [[function sets]]. This exists, by the discussion there, as soon as the [[foundations|foundational]] [[axioms]] are strong enough, for instance as soon as there are [[power objects]], which is the special case of a function set into the 2-element set.

\hypertarget{in_simplicial_sets}{}\subsubsection*{{In simplicial sets}}\label{in_simplicial_sets}

In the category [[sSet]] of [[simplicial sets]], the internal hom between two [[simplicial sets]] $X,Y$ is given by the formula

\begin{displaymath}
[X,Y]_n = Hom_{sSet}(X\times \Delta[n],Y)
  \,,
\end{displaymath}
where $\Delta[n]$ is the simplicial [[n-simplex]]. This $[X,Y] \in sSet$ is also called the \emph{[[function complex]]} between $X$ and $Y$.

Since $sSet \simeq PSh(\Delta)$ is the [[category of presheaves]] over the [[simplex category]], this is a special case of internal homs in sheaf toposes, discussed \hyperlink{InASheafTopos}{below}.

\hypertarget{InASheafTopos}{}\subsubsection*{{In a sheaf topos or $(\infty,1)$-sheaf $(\infty,1)$-topos}}\label{InASheafTopos}

Let $C$ be a [[site]]. Let $\mathbf{H} = Sh(C)$ be the [[sheaf topos]] over $C$ or in fact the [[(∞,1)-sheaf (∞,1)-topos]]. We discuss the internal hom of this regard as a [[cartesian monoidal category]]/[[cartesian monoidal (∞,1)-category]].

\begin{prop}
\label{}\hypertarget{}{}
The sheaf topos $\mathbf{H}$ is a [[cartesian closed category]] / [[cartesian closed (∞,1)-category]]. In fact it is a [[locally cartesian closed category]] / [[locally cartesian closed (∞,1)-category]].

\end{prop}
Hence the internal hom exist.

=--

\begin{prop}
\label{InternalHomInSheaves}\hypertarget{InternalHomInSheaves}{}
For $X, Y \in \mathbf{H}$ two [[objects]], the internal hom-object

\begin{displaymath}
[X,Y] \in \mathbf{H}
\end{displaymath}
is the [[sheaf]]/[[(∞,1)-sheaf]] given by the assignment

\begin{displaymath}
[X,Y] : U \mapsto \mathbf{H}(U \times X, Y)
  \,,
\end{displaymath}
for all objects $U \in C$ which on the right we regard under the [[Yoneda embedding]]/[[(∞,1)-Yoneda lemma|∞-Yoneda embedding]] $U \in C \stackrel{Yoneda}{\hookrightarrow} \mathbf{H}$.

Here

\begin{itemize}%
\item $U \times X \in \mathbf{H}$ is the [[cartesian product]] if $U$ with $X$


\item $\mathbf{H}(-,-)$ is the [[hom set]]-[[functor]] / [[derived hom-space|hom space]]-[[(∞,1)-functor]] of $\mathbf{H}$.



\end{itemize}
\end{prop}
See also at \emph{[[closed monoidal structure on presheaves]]}.

\begin{proof}
By the [[Yoneda lemma]]/[[(∞,1)-Yoneda lemma]] we have [[natural equivalences]]

\begin{displaymath}
[X,Y](U) \simeq \mathbf{H}(U , [X,Y])
\end{displaymath}
and by the defining $((-)\times X \vdash [X,-])$[[adjunction]] this is naturally equivalent to

\begin{displaymath}
\cdots \simeq \mathbf{H}(U \times X, Y)
  \,.
\end{displaymath}
\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
In the ([[homotopy type theory|homotopy]]-)[[type theory]] [[syntax]] of the [[internal language]] of $\mathbf{H}$ the internal hom $[X, Y] \in \mathbf{H}$ is the [[categorical semantics]] of the [[function type]]

\begin{displaymath}
\vdash (X \to Y) : Type
  \,.
\end{displaymath}
\end{remark}
\begin{prop}
\label{EvaluationOfInternalFunctionsInSheafTopos}\hypertarget{EvaluationOfInternalFunctionsInSheafTopos}{}
For $X, Y \in \mathbf{H}$, the [[evaluation map]], def. \ref{EvalMap},

\begin{displaymath}
eval_{X,Y} : [X,Y] \times X \to Y
\end{displaymath}
is the morphism of sheaves which over each $U \in C$ sends a morphism of sheaves $\theta : \mathbf{H}(-,U) \times X(-) \to Y(-)$ (which is the first component by prop. \ref{InternalHomInSheaves}) and an $x \in \mathbf{H}(U,X)$ to

\begin{displaymath}
eval_{X,Y}(U) : (\theta, x) \mapsto \theta_U(id_U, x) \in Y(U)
  \,.
\end{displaymath}
\end{prop}
See (\hyperlink{MacLane-Moerdijk}{MacLane-Moerdijk, p. 46}).

\begin{prop}
\label{CompositionOfInternalFunctionsInSheafTopos}\hypertarget{CompositionOfInternalFunctionsInSheafTopos}{}
For $X, Y, Z \in \mathbf{H}$ three [[objects]] of $\mathbf{H}$, the canonical [[composition]] [[morphism]], def. \ref{CompositionMorphism},

\begin{displaymath}
\circ_{X,Y,Z} : 
  [Y, Z] \times [X, Y] \to [X, Z]
\end{displaymath}
is given by the morphism of [[presheaves]]/[[(∞,1)-presheaves]] whose component over $U \in C$ is the morphism of [[sets]]/[[∞-groupoids]]

\begin{displaymath}
\circ_{X,Y,Z}(U)
  : 
  \mathbf{H}(U \times X, Y)
  \times
  \mathbf{H}(U \times Y, Z)
  \to 
  \mathbf{H}(U \times X, Z)
\end{displaymath}
which sends a pair $(f : U \times X \to Y, g : U \times Y \to Z)$ to the composite

\begin{displaymath}
\circ_{X,Y,Z}(U)(f,g)
  = 
  U \times X 
  \stackrel{(\Delta_U, id_X)}{\to}
  U \times U \times X
  \stackrel{(id_U, f)}{\to}
  U \times Y 
  \stackrel{g}{\to}
  Z
  \,,
\end{displaymath}
where $\Delta_U : U \to U \times U$ is the [[diagonal]] morphism on $U$.

\end{prop}
\begin{proof}
By definition \ref{CompositionMorphism} the morphism is the [[adjunct]] of the double [[evaluation map]]

\begin{displaymath}
[Y,Z] \times [X,Y] \times X \to Z
  \,.
\end{displaymath}
Since the [[cartesian product]] of two sheaves $A, B \in \mathbf{H}$ is computed objectwise

\begin{displaymath}
A \times B : U \mapsto A(U) \times B(U)
\end{displaymath}
it follows that over each $U \in C$ this double evaluation map is the morphism of sets/$\infty$-groupoids

\begin{displaymath}
[Y,Z](U) \times [X,Y](U) \times X(U) \to Z(U)
\end{displaymath}
hence by prop. \ref{InternalHomInSheaves}

\begin{displaymath}
\mathbf{H}(U \times Y, Z) \times \mathbf{H}(U \times X, Y) \times 
  \mathbf{H}(U,X) \to \mathbf{H}(U,Z)
  \,,
\end{displaymath}
where now by prop. $\backslash$ref this is the external evaluation.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
Intuitively this says that the composite of a $U$-parameterized family of maps $\{f(u) :  X \to Y| u \in U\}$ with a $U$-parameterized family of maps ${g(u) : Y \to Z| u \in U}$ is the $U$-family given by the parameter-wise composite $\{g(u)\circ f(u) | u \in U\}$.

\end{remark}
\begin{example}
\label{InternalAutomorphismGroup}\hypertarget{InternalAutomorphismGroup}{}
The internal [[automorphism group]]/[[automorphism ∞-group]] of an [[object]] $X \in \mathbf{H}$ is the [[subobject]]

\begin{displaymath}
\mathbf{Aut}(X) \hookrightarrow [X,X]
\end{displaymath}
of the internal hom which is maximal subject to the property that the composition of prop. \ref{CompositionOfInternalFunctionsInSheafTopos} becomes invertible.

The ([[homotopy type theory|homotopy]]-)[[type theory]] [[syntax]] for this is given by the [[type]] of [[equivalences in homotopy type theory]]

\begin{displaymath}
\vdash (X \stackrel{\simeq}{\to} X) : Type
  \,.
\end{displaymath}
\end{example}
\hypertarget{ExampleInSliceCategories}{}\subsubsection*{{In slice categories}}\label{ExampleInSliceCategories}

Let $\mathbf{H}$ be a [[locally cartesian closed category]]. This means that for each object $X \in \mathbf{H}$ the [[slice category]] $\mathbf{H}_{/X}$ is a [[cartesian closed category]]. The [[product]] in the slice is given by the [[fiber product]] over $X$ computed in $\mathbf{H}$. Fairly detailed discussion of constructions of the internal hom in such slices $\mathbf{H}_{/X}$ is at \emph{\href{locally%20cartesian%20closed%20category#EquivalentCharacterizations}{locally cartesian closed category -- cartesian closure in terms of base change and dependent product}}.

We record some further properties

\begin{prop}
\label{InverseImageBaseChangeIsCartesianClosed}\hypertarget{InverseImageBaseChangeIsCartesianClosed}{}
For $\mathcal{C}$ a [[locally cartesian closed category]] and $f \colon X \to Y$ any morphism in $\mathcal{C}$, the [[inverse image]] $f^*$ of the corresponding [[base change]] [[adjunction]]

\begin{displaymath}
\mathcal{C}_{/X}
   \stackrel{\overset{\sum_f}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{\prod_f}{\to}}}
  \mathcal{C}_{/Y}
\end{displaymath}
is a [[cartesian closed functor]].

\end{prop}
This is discussed in more detail at \emph{\href{cartesian+closed+functor#Examples}{cartesian closed functor -- Examples}}.

So for $A,B \in \mathcal{C}_{/Y}$ we have [[isomorphisms]]

\begin{displaymath}
f^* \left[A,B\right] \stackrel{\simeq}{\to} \left[f^* A , f^* B\right]
\end{displaymath}
between the image of the internal hom under $f^*$ and the internal hom of the images of $A$ and $B$ separately.

\begin{prop}
\label{MorphismFromDepProductOfFuncTypeToFuncTypeOfDepSum}\hypertarget{MorphismFromDepProductOfFuncTypeToFuncTypeOfDepSum}{}
For $\mathbf{H}$ a [[locally cartesian closed category]], $f \colon X \to Y$ any [[morphism]], and $A, B \in \mathbf{H}_{/X}$ two objects in the slice over $X$, there is a natural morphism (not in general an isomorphism)

\begin{displaymath}
\prod_f \left[A,B \right] \to \left[ \sum_f A, \sum_f B\right]
  \,.
\end{displaymath}
\end{prop}
Here are two ways to get this morphism:

\begin{proof}
For any object $U \in \mathbf{H}_{/Y}$ we have a canonical morphism of [[hom sets]]

\begin{displaymath}
\begin{aligned}
     \mathbf{H}_{/Y}( U, \prod_f [A,B] )
     & \simeq
     \mathbf{H}_{/X}( f^* U, [A,B] )
     \\
     & \simeq
     \mathbf{H}_{/X}(f^* U \times A, B)
     \\
     & \stackrel{}{\to}
    \mathbf{H}_{/Y}( \sum_f( f^* U \times A ), \sum_f B )
    \\
     & \stackrel{Frob.Rec.}{\simeq}
     \mathbf{H}_{/Y}( U \times \sum_f A , \sum_f B )
     \\
     & \simeq
    \mathbf{H}_{/Y}(U, [\sum_f A , \sum_f B])
  \end{aligned}
\end{displaymath}
where the first and the last steps use [[adjunction]] properties, where the morphism in the middle is the component of the [[dependent sum]] functor, and where ``Frob.Rec.'' is [[Frobenius reciprocity]].

Since this is [[natural transformation|natural]] in $U$, the [[Yoneda lemma]] implies the claimed morphism.

\end{proof}
\begin{proof}
There is the composite morphism

\begin{displaymath}
\left(f^\ast \prod_f [A, B]\right) \times A \stackrel{counit \times id_A}{\to} [A, B] \times A \stackrel{eval}{\to} B \stackrel{unit}{\to} f^\ast \sum_f B
\end{displaymath}
of the [[unit of an adjunction|adjunction (co)units]] and the [[evaluation map]] of the internal hom. Its hom-[[adjunct]] is

\begin{displaymath}
A \to [f^\ast \prod_f [A, B], f^\ast \sum_f B] \cong f^\ast [\prod_f [A, B], \sum_f B]
  \,,
\end{displaymath}
using prop. \ref{InverseImageBaseChangeIsCartesianClosed} on the right. The hom-adjunct of that in turn is

\begin{displaymath}
\sum_f A \to [\prod_f [A, B], \sum_f B]
\end{displaymath}
and by symmetry the morphism that we are after:

\begin{displaymath}
\prod_f [A, B] \to [\sum_f A, \sum_f B]
  \,.
\end{displaymath}
\end{proof}
\begin{remark}
\label{RememberingTopMorphismInHomInSlice}\hypertarget{RememberingTopMorphismInHomInSlice}{}
If $Y$ is the [[terminal object]] (for simplicity), then the morphism of prop. \ref{MorphismFromDepProductOfFuncTypeToFuncTypeOfDepSum} can be understood as follows: a [[global element]] of the [[dependent product]] $\prod_f [A,B]$ is given by a [[commuting diagram]] in $\mathbf{H}$ of the form

\begin{displaymath}
\itexarray{
    \sum_f A &&\to&& \sum_f B
    \\
     & \searrow && \swarrow
     \\
     && X
  }
  \,.
\end{displaymath}
The map in prop. \ref{MorphismFromDepProductOfFuncTypeToFuncTypeOfDepSum} picks out the top horizontal morphism in this diagram.

\end{remark}
\hypertarget{for_smooth_spaces_and_smooth_groupoids}{}\subsubsection*{{For smooth spaces and smooth $\infty$-groupoids}}\label{for_smooth_spaces_and_smooth_groupoids}

Consider the [[site]] $C =$ [[SmthMfd]] of [[smooth manifolds]] (and the [[open cover]] [[coverage]]) or equivalently over the [[dense subsite]] [[CartSp]] of [[Cartesian spaces]] and [[smooth functions]] between these.

The [[sheaf topos]]/[[(∞,1)-sheaf (∞,1)-topos]] $\mathbf{H} = Sh(C)$ is that of [[smooth spaces]]/[[smooth ∞-groupoids]]. So the discussion of internal homs here is a special case of the above discussion \emph{\hyperlink{InASheafTopos}{In a sheaf topos}}.

\begin{example}
\label{}\hypertarget{}{}
For $X , Y \in SmthMfd \hookrightarrow \mathbf{H}$ two [[smooth manifolds]], the [[internal hom]] $[X,Y] \in \mathbf{H}$ is the [[mapping space]] between them regarded as a [[diffeological space]].

See at \emph{[[manifold structure of mapping spaces]]} for when this internal hom is [[representable functor|representable]] again by a [[smooth manifold]].

\end{example}
\begin{example}
\label{}\hypertarget{}{}
For $X \in SmthMfd \hookrightarrow \mathbf{H}$ the internal automorphism group, example \ref{InternalAutomorphismGroup}, of $X$ is the [[diffeomorphism group]] of $X$, regarded as a [[diffeological space|diffeological]] group

\begin{displaymath}
\mathbf{Aut}(X) = \mathbf{Diff}(X)
  \,.
\end{displaymath}
\end{example}
\hypertarget{for_chain_complexes}{}\subsubsection*{{For chain complexes}}\label{for_chain_complexes}

\begin{itemize}%
\item [[internal hom of chain complexes]]

\end{itemize}
\hypertarget{for_super_vector_spaces}{}\subsubsection*{{For super vector spaces}}\label{for_super_vector_spaces}

The category $sVect$ of [[super vector spaces]] is the category of $\mathbb{Z}/2$-[[graded vector spaces]]. Thus, its objects are pairs of vector spaces $(V_+,V_-)$, with $V_+$ called the \emph{even} part and $V_-$ the \emph{odd} part. The morphisms in $sVect$ are likewise pairs of linear maps, i.e. we define $sVect$ to be $Vect \times Vect = Vect^{\mathbb{Z}/2}$, as usual for any sort of graded object. With this definition of the category $sVect$, we capture the concepts of superalgebra and so on in succinct categorical terms.

Because the morphisms in $sVect$ send even things to even things and odd things to odd things, they are sometimes called \emph{even} linear maps, and one may write

\begin{displaymath}
sVect(V, W) = Even Lin(V,W).
\end{displaymath}
Note that $sVect$ is [[enriched category|enriched]] over $Vect$, i.e. these hom-sets are vector spaces.

Occasionally, however, one does need to refer to the \emph{odd} linear maps, which send even things to odd things and odd things to even things. That is, an odd linear map $V\to W$ is a pair of linear maps $V_+ \to W_-$ and $V_-\to W_+$. The internal-hom in $sVect$ allows us to capture these as well: it is the following super vector space:

\begin{displaymath}
[V,W]_+ = Even Lin(V,W) \qquad
  [V,W]_- = Odd Lin(V,W).
\end{displaymath}
With this definition, $sVect$ becomes a [[closed monoidal category]].

We can equivalently regard a super vector spaces $(V_+,V_-)$ as being the [[direct sum]] vector space $V_+ \oplus V_-$ equipped with this direct sum decomposition. If we view the internal-hom $[V,W]$ in this way as well, then we have

\begin{displaymath}
[V, W] = Even Lin(V,W) \oplus Odd Lin(V,W) = Lin(V,W).
\end{displaymath}
In other words, any linear map between these ``summed'' super vector spaces decomposes uniquely as the sum of an even linear map and an odd one.

\hypertarget{for_banach_spaces}{}\subsubsection*{{For Banach spaces}}\label{for_banach_spaces}

A similar thing happens in the category $Ban$ of [[Banach spaces]] and [[short linear operators]]. The external hom consists of only the \emph{short} linear maps (those bounded by $1$):

\begin{displaymath}
Ban(V,W) = \{ f\colon Lin(V,W) \;|\; {\|f\|} \leq 1 \} .
\end{displaymath}
This definition of morphism recovers the most specific notion of [[isomorphism]] of Banach spaces, as well as defining the [[product]] and [[coproduct]] as the [[direct sum]] completed with $p = \infty$ or $p = 1$ respectively.

But the internal hom is the Banach space of \emph{all} bounded linear maps:

\begin{displaymath}
[V,W] = \{ f\colon Lin(V,W) \;|\; {\|f\|} \lt \infty \} .
\end{displaymath}
This is a Banach space and makes $Ban$ into a [[closed category]].

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

\begin{itemize}%
\item [[hom-set]], [[hom-object]]


\item [[hom-functor]]


\item [[enriched hom-functor]]


\item [[derived hom-functor]]


\item [[function type]]

\begin{itemize}%
\item [[implication]], [[linear implication]]

\end{itemize}

\item [[power object]], [[exponential object]]


\item [[function monad]]


\item [[exponential law for spaces]]


\item [[closed category]], [[cartesian closed category]]


\item [[strong adjoint functor]]


\item [[mapping stack]]


\item [[space of sections]]


\item [[pointed mapping space]]


\item [[distributions are the smooth linear functionals]]


\item [[Sullivan model of mapping space]]



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

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

\end{itemize}
A discussion query (R. Brown, T. Bartels, M. Shulman) about internal hom is at $n$Forum \href{http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=3372&Focus=27648}{here}.

[[!redirects internal homs]] [[!redirects inner hom]] [[!redirects inner homs]] [[!redirects internal-hom]] [[!redirects internal-homs]]

[[!redirects mapping space]] [[!redirects mapping spaces]]

[[!redirects exponential law]] [[!redirects exponential laws]]

[[!redirects internal hom-functor]] [[!redirects internal hom-functors]] [[!redirects internal hom functor]] [[!redirects internal hom functors]]



\end{document}