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\section*{integral transforms on sheaves}

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

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory}

[[!include (infinity,1)-topos - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{linear_bases}{Linear bases}\dotfill \pageref*{linear_bases} \linebreak
\noindent\hyperlink{homspaces}{Hom-spaces}\dotfill \pageref*{homspaces} \linebreak
\noindent\hyperlink{tensor_products}{Tensor products}\dotfill \pageref*{tensor_products} \linebreak
\noindent\hyperlink{FunctionSpaces}{Function spaces}\dotfill \pageref*{FunctionSpaces} \linebreak
\noindent\hyperlink{products_of_function_objects}{Products of function objects}\dotfill \pageref*{products_of_function_objects} \linebreak
\noindent\hyperlink{fiber_integration}{Fiber integration}\dotfill \pageref*{fiber_integration} \linebreak
\noindent\hyperlink{integral_transforms}{Integral transforms}\dotfill \pageref*{integral_transforms} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

There is a sense in which a [[sheaf]] $F$ is like a [[categorification]] of a [[function]]: We may think of the [[stalk]]-map from [[point of a topos|topos point]]s to [[set]]s $(x^* \dashv x_*) \mapsto F(x) := x^* F \in Set$ under [[decategorification]] as a [[cardinality]]-valued [[function]].

Under this interpretation, many constructions in [[category theory]] have analogs in [[linear algebra]]: for instance products of numbers correspond to categorical [[product]]s (more generally to [[limit]]s) and addition of numbers to [[coproduct]]s (more generally to [[colimit]]s). Accordingly a [[colimit]]-preserving [[functor]] between [[sheaf topos]]es is analogous to a [[linear map]] or to a [[distribution]]: one also speaks of [[Lawvere distribution]]s.

This [[categorification]] of [[linear algebra]] becomes even better behaved if we pass all the way to [[(∞,1)-sheaf (∞,1)-topos]]es. Under [[∞-groupoid cardinality]] their [[stalk]]s take values also in [[integer]]s, in [[rational number]]s, and in [[real number]]s. See also the discussion at [[Goodwillie calculus]].

A [[span]] of [[base change geometric morphism]]s between toposes behaves under this interpretation like the [[linear map]] given by a [[matrix]]. Such categorified [[integral transform]]s turn out to be of considerable interest in their own right: they include operations such as the [[Fourier-Mukai transform]] which categorifies the [[Fourier transform]].

These analogies have been noticed and exploited at various places in the literature. See for instance the entries [[groupoidification]] or [[geometric ∞-function theory]]. Here we try to give a general abstract [[(∞,1)-topos theory|(∞,1)-topos theoretic]] description with examples from ordinary [[topos theory]] to motivate the constructions.

\hypertarget{linear_bases}{}\subsection*{{Linear bases}}\label{linear_bases}

Every [[(∞,1)-topos]] is a [[locally presentable (∞,1)-category]]. More generally we may think of arbitrary locally presentable [[(∞,1)-categories]] as being analogous to [[vector space]]s of [[linear functional]]s.

\begin{uprop}
Every [[locally presentable (∞,1)-category]] is a [[reflective sub-(∞,1)-category]] of an [[(∞,1)-category of (∞,1)-presheaves]].

\end{uprop}
See [[locally presentable (∞,1)-category]] for details.

\begin{uremark}
For $C$ a small [[(∞,1)-category]] the [[(∞,1)-category of (∞,1)-presheaves]]

\begin{displaymath}
\hat C := Func(C^{op}, \infty Grpd)
\end{displaymath}
is the [[free cocompletion|free (∞,1)-cocompletion]] of $C$, hence the free completion under [[(∞,1)-colimit]]s. Under the interpretation of colimits as sums, this means that it is analogous to the vector spaces \emph{spanned} by the [[basis]] $C$.

Accordingly an arbitrary locally presentable $(\infty,1)$-category is analogous in this sense to a sub-space of a vector space spanned by a basis.

\end{uremark}
\begin{uprop}
For $\hat C, \hat D$ two [[(∞,1)-categories of (∞,1)-presheaves]], a morphism $\hat C \to \hat D$ in [[Pr(∞,1)Cat]] is equivalently a [[(∞,1)-profunctor]] $C  &#8696; D$.

\end{uprop}
See [[profunctor]] for details.

\hypertarget{homspaces}{}\subsection*{{Hom-spaces}}\label{homspaces}

\begin{uprop}
For $C, D \in$ [[Pr(∞,1)Cat]] we have that $Func^L(C,D)$ is itself locally presentable.

\end{uprop}
See [[Pr(∞,1)Cat]] for details.

\begin{uremark}
This means that to the extent that we may think of $C, D$ as analogous to vector spaces, also the space of linear maps between them is analogous to a vector space.

\end{uremark}
\hypertarget{tensor_products}{}\subsection*{{Tensor products}}\label{tensor_products}

\begin{uprop}
For $C$ and $D$ two [[locally presentable (∞,1)-categories]] there is locally presentable $(\infty,1)$-category $C \otimes D$ and an [[(∞,1)-functor]]

\begin{displaymath}
C \times D \to C \otimes D
\end{displaymath}
which is [[universal property|universal]] with respect to the property that it preserves [[(∞,1)-colimit]]s in both arguments.

\end{uprop}
\begin{uremark}
This means that in as far as $C, D \in$ [[Pr(∞,1)Cat]] are analogous to vector spaces, $C \otimes D$ is analogous to their [[tensor product]].

\end{uremark}
\hypertarget{FunctionSpaces}{}\subsection*{{Function spaces}}\label{FunctionSpaces}

We consider from now on some fixed ambient [[(∞,1)-topos]] $\mathbf{H}$.

Notice that for each [[object]] $X \in \mathbf{H}$ the [[over-(∞,1)-topos]] $\mathbf{H}/X$ is the [[little topos]] of $(\infty,1)$-sheaves on $X$. So to the extent that we think of these as \textbf{function objects}, and of locally presentable $(\infty,1)$-categories as linear spaces, we may think of $\mathbf{H}/X$ as the $\infty$-vector space of $\infty$-functions on $X$

\begin{uremark}
The [[over-(∞,1)-topos]]es $\mathbf{H}/X$ sit by an [[etale geometric morphism]] over $\mathbf{H}$ and are characterized up to equivalence by this property.

Moreover, we have an equivalence of the ambient $(\infty,1)$-topos $\mathbf{H}$ with the $(\infty,1)$-category of [[etale geometric morphism]]s into it.

\begin{displaymath}
((\infty,1)Topos/\mathbf{H})_{et} \simeq
  \mathbf{H}
  \,.
\end{displaymath}
\end{uremark}
\begin{uexample}
Let $\mathbf{H} =$ [[FinSet]] be the ordinary [[topos]] of [[finite set]]s. Then for $X \in FinSet$ a finite set, a \emph{function object} on $X$ is a morphism $\psi : \Psi \to X$ of sets. Under the [[cardinality]] [[decategorification]]

\begin{displaymath}
|-| : FinSet \to \mathbb{N}
\end{displaymath}
we think of this as the [[function]]

\begin{displaymath}
|\psi| : X \to \mathbb{N}
\end{displaymath}
given by

\begin{displaymath}
x \mapsto |\Psi_x|
  \,,
\end{displaymath}
where $\Psi_x \in FinSet$ is the [[fiber]] of $\psi$ over $X$.

\end{uexample}
\begin{uexample}
Let $\mathbf{H} =$ [[∞Grpd]]. By the [[(∞,1)-Grothendieck construction]] we have for $X \in \infty Grpd$ an [[∞-groupoid]] an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
\infty Grpd/X \simeq PSh_{(\infty,1)}(X) \simeq  Func_{(\infty,1)}(X,\infty Grpd)
\end{displaymath}
of the [[over-(∞,1)-category]] of all $\infty$-groupoids over $X$ with the [[(∞,1)-category of (∞,1)-presheaves]] on $X$. And since the $\infty$-groupoid $C$ is equivalent to its [[opposite (∞,1)-category]] this is also equivalent to the [[(∞,1)-category of (∞,1)-functors]] from $C$ to [[∞Grpd]].

\end{uexample}
\hypertarget{products_of_function_objects}{}\subsection*{{Products of function objects}}\label{products_of_function_objects}

For $\psi : \Psi \to X$ and $\phi : \Phi \to X$ in $\mathbf{H}/X$ two function objects on $X$, their [[product]] $\psi \times \phi$ in $\mathbf{H}/X$ we call the \textbf{product of function objects}.

This is computed in $\mathbf{H}$ as the [[fiber product]]

\begin{displaymath}
\psi \times^{\mathbf{H}/X} \phi = \Psi \times^{\mathbf{H}}_X \Phi
\end{displaymath}
and the morphism down to $X$ is the evident projection

\begin{displaymath}
\itexarray{
    && \Psi \times_{X}^{\mathbf{H}} \Phi
    \\
    & \swarrow && \searrow
    \\
    \Psi &&\downarrow^{\psi \times^{\mathbf{H}/X} \phi}&& \Phi
    \\
    & {}_{\mathllap{\psi}}\searrow && \swarrow_{\mathrlap{\phi}}
    \\
    && X
  }
  \,.
\end{displaymath}
\begin{uexample}
In $\mathbf{H} =$ [[FinSet]] we have that the $\mathbb{N}$-valued function underlying the product function object is the usual pointwise product of functions

\begin{displaymath}
|\psi \cdot \phi| : x \mapsto |\psi|(x) \cdot |\phi|(x)
  \,.
\end{displaymath}
\end{uexample}
\hypertarget{fiber_integration}{}\subsection*{{Fiber integration}}\label{fiber_integration}

For every [[morphism]] $v : X \to Y$ in the ambient [[(∞,1)-topos]] $\mathbf{H}$ there is the corresponding [[base change geometric morphism]]

\begin{displaymath}
(v_! \dashv v^* \dashv v_*) 
    : 
  \mathbf{H}/X
   \stackrel{\overset{v_!}{\to}}{\stackrel{\overset{v^*}{\leftarrow}}{\underset{v_*}{\to}}}
   \mathbf{H}/Y
\end{displaymath}
between the corresponding [[over-(∞,1)-topos]]es. Here $v_!$ acts simply by postcomposition with $v$:

\begin{displaymath}
v_! : (\Psi \stackrel{\psi}{\to} X) \mapsto (\Psi \stackrel{\psi}{\to} X \stackrel{v}{\to} Y)
\end{displaymath}
while $v^*$ acts by [[(∞,1)-pullback]] along $v$:

\begin{displaymath}
v^* : (\Phi \stackrel{\phi}{\to} Y) \mapsto (X \times_Y \Phi)
  \,.
\end{displaymath}
There is a further [[right adjoint]] $v_*$. For the present purpose the relevance of its existence is that it implies that both $v_!$ as well as $v^*$ are [[left adjoint]]s and hence both preserve [[(∞,1)-colimit]]s. Therefore these are morphism in [[Pr(∞,1)Cat]] and hence behave like linear maps on our function spaces $\mathbf{H}/X$ and $\mathbf{H}/Y$.

When we think of [[base change]] in the context of linear algebra on sheaves, we shall write $\int_{X/Y}  := v_!$

\begin{displaymath}
(\int_{X/Y} \dashv v^*) : 
  \mathbf{H}/X
  \stackrel{\overset{\int_{X/Y}}{\to}}{\underset{v^*}{\leftarrow}}
  \mathbf{H}/Y
\end{displaymath}
and call $\int_{X/Y} \psi$ the \textbf{[[fiber integration]]} of $F$ over the fibers of $v$. In particular when $Y = *$ is the [[terminal object]] we write simply

\begin{displaymath}
\int_X \psi \in \mathbf{H}
\end{displaymath}
for the \textbf{integral} of $\psi$ with values in the ambient $(\infty,1)$-topos. (See also the notation for [[Lawvere distribution]]s).

\begin{uexample}
Consider the ordinary [[topos]] $\mathbf{H} =$ [[FinSet]] and for $X \in \mathbf{H}$ any set the unique [[morphism]] $v : X \to *$ to the [[terminal object]].

For $\psi : \Psi \to X$ a function object with underlying function $\psi : x \mapsto |\Psi_x|$ we have that the integral

\begin{displaymath}
\int_X \psi : \Psi \to *
\end{displaymath}
has as underlying function the constant

\begin{displaymath}
|\int_X \psi| = \sum_{x \in X} |\psi|(x)
  \,.
\end{displaymath}
\end{uexample}
\hypertarget{integral_transforms}{}\subsection*{{Integral transforms}}\label{integral_transforms}

If we are given an oriented [[span]] or [[correspondence]]

\begin{displaymath}
\left(
  \itexarray{  
     && A
     \\
     & {}^{\mathllap{i}}\swarrow && \searrow^{\mathrlap{o}}
     \\
     X &&&& Y
  }
  \right)
\end{displaymath}
in $\mathbf{H}$ it induces by composition of pullback and fiber integration operations a colimit-preserving $(\infty,1)$-functor

\begin{displaymath}
\underline{A}
  :
  \mathbf{H}/X
   \stackrel{i^*}{\to}
  \mathbf{H}/A
   \stackrel{\int_{A/Y}}{\to}
  \mathbf{H}/Y
  \,.
\end{displaymath}
We may always factor $(i,o)$ through the [[(∞,1)-limit|(∞,1)-product]]

\begin{displaymath}
\left(
  \itexarray{
     && A
     \\
     && \downarrow^{\mathrlap{(i,o)}}
     \\
     && X \times Y
     \\
     & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}}
     \\
     X &&&& Y
  }
  \right)
  \,.
\end{displaymath}
We call the function object

\begin{displaymath}
((i,o) : A \to X \times Y) \in \mathbf{H}/(X\times Y)
\end{displaymath}
on $X \times Y$ the \textbf{integral kernel} of $\underline{A}$.

\begin{ulemma}
We have the \textbf{pull-tensor-push formula} for $\underline{A}$:

\begin{displaymath}
\underline{A} F = \int_{A/Y} i^* F =  (p_2)_!(A \times (p_1^* F) )
  \,.
\end{displaymath}
\end{ulemma}
\begin{proof}
This follows from the  for pullbacks in $\mathbf{H}$:

\begin{displaymath}
\itexarray{
    i^* \Psi &\to& p_1^* \Psi &\to& \Psi
    \\
    \downarrow && \downarrow && \downarrow^{\mathrlap{\psi}}
    \\
    A &\stackrel{(i,o)}{\to}& X \times Y &\stackrel{p_1}{\to}& X
  }
  \,.
\end{displaymath}
\end{proof}
\begin{uremark}
By the \hyperlink{EtaleIdentificationRemark}{above remark} on [[etale geometric morphism]]s we have that we can recover the span $X \stackrel{i}{\leftarrow} A \stackrel{o}{\to} Y$ in $\mathbf{H}$ from the span

\begin{displaymath}
\itexarray{
     \mathbf{H}/X 
    &\stackrel{\overset{i^*}{\to}}{\underset{i_*}{\leftarrow}}&
      \mathbf{H}/A
    &\stackrel{\overset{o^*}{\leftarrow}}{\underset{o_*}{\to}}&
    \mathbf{H}/Y
    \\
    & \searrow\nwarrow & \downarrow\uparrow & \swarrow\nearrow
    \\
    && \mathbf{H}
  }
\end{displaymath}
in $((\infty,1)Topos/\mathbf{H})_{et}$.

\end{uremark}
\begin{uexample}
In $\mathbf{H} =$ [[FinSet]] we have that $(|A_{x,y}|)$ is a $|X|$-by-$|Y|$-[[matrix]] with entries in [[natural number]]s and the function

\begin{displaymath}
|A \psi | : y \mapsto | (i^* \Psi)_y | = \sum_{x \in X} |A_{x,y}| \cdot |\psi|(x)
\end{displaymath}
is the result of applying the familiar [[linear map]] given by usual [[matrix calculus]] on $|\psi|$.

\end{uexample}
\begin{uexample}
In the case $\mathbf{H} =$ [[∞Grpd]] we have -- as in the \hyperlink{PresheavesOnGroupoids}{above example} -- by the [[(∞,1)-Grothendieck construction]] an equivalence

\begin{displaymath}
\infty Grpd / (X \times Y) \simeq PSh_{(\infty,1)}(X \times Y)
  \,.
\end{displaymath}
Since the $\infty$-groupoid $Y$ is equivalent to its [[opposite (∞,1)-category]] this may also be written as

\begin{displaymath}
\infty Grpd / (X \times Y) \simeq  Func_{(\infty,1)}(X \times Y^{op}, \infty Grpd)
  \,.
\end{displaymath}
The objects on the right we may again think of as $(\infty,1)$-[[profunctor]]s $X &#x21F8; Y$. So in particular the kernel $(A \to X \times Y) \in \infty Grpd/(X \times Y)$ is under this equivalence on the right hand identified with an $(\infty,1)$-profunctor

\begin{displaymath}
\tilde A : X &#x21F8; Y
  \,.
\end{displaymath}
\end{uexample}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[integral transform]]

\begin{itemize}%
\item [[Fourier-Mukai transform]], [[Hecke transform]]

\end{itemize}

\item [[polynomial functor]]


\item [[pure motives]], [[motives in physics]], [[motivic quantization]]


\item [[path integral as a pull-push transform]]



\end{itemize}


\end{document}