# nLab integral transforms on sheaves

### Context

#### Higher category theory

higher category theory

## 1-categorical presentations

#### $\left(\infty ,1\right)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Idea

There is a sense in which a sheaf $F$ is like a categorification of a function: the stalk-map from topos points to sets $\left({x}^{*}⊣{x}_{*}\right)↦F\left(x\right):={x}^{*}F\in \mathrm{Set}$ we may think of 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 products (more generally to limits) and addition of numbers to coproducts (more generally to colimits). Accordingly a colimit-preserving functor between sheaf toposes is analogous to a linear map or to a distribution: one also speaks of Lawvere distributions.

This categorification of linear algebra becomes even better behaved if we pass all the way to (∞,1)-sheaf (∞,1)-toposes. Under ∞-groupoid cardinality their stalks take values also in integers, in rational numbers, and in real numbers. See also the discussion at Goodwillie calculus.

A span of base change geometric morphisms between toposes behaves under this interpretation like the linear map given by a matrix. Such categorified integral transforms 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 theoretic description with examples from ordinary topos theory to motivate the constructions.

## 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 spaces of linear functionals.

###### Proposition

See locally presentable (∞,1)-category for details.

###### Remark

For $C$ a small (∞,1)-category the (∞,1)-category of (∞,1)-presheaves

$\stackrel{^}{C}:=\mathrm{Func}\left({C}^{\mathrm{op}},\infty \mathrm{Grpd}\right)$\hat C := Func(C^{op}, \infty Grpd)

is the free (∞,1)-cocompletion of $X$, hence the free completion under (∞,1)-colimits. Under the interpretation of colimits as sums, this means that it is analogous to the vector spaces spanned by the basis $C$.

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

###### Proposition

For $\stackrel{^}{C},\stackrel{^}{D}$ two (∞,1)-categories of (∞,1)-presheaves, a morphism $\stackrel{^}{C}\to \stackrel{^}{D}$ in Pr(∞,1)Cat is equivalently a profunctor $C⇸D$.

See profunctor for details.

## Hom-spaces

###### Proposition

For $C,D\in$ Pr(∞,1)Cat we have that ${\mathrm{Func}}^{L}\left(C,D\right)$ is itself locally presentable.

See Pr(∞,1)Cat for details.

###### Remark

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.

## Tensor products

###### Fact

For $C$ and $D$ two locally presentable (∞,1)-categories there is locally presentable $\left(\infty ,1\right)$-category $C\otimes D$ and an (∞,1)-functor

$C×D\to C\otimes D$C \times D \to C \otimes D

which is universal with respect to the property that it preserves (∞,1)-colimits in both arguments.

###### Remark

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.

## Function spaces

We consider from now on some fixed ambient (∞,1)-topos $H$.

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

###### Remark

The over-(∞,1)-toposes $H/X$ sit by an etale geometric morphism over $H$ and are characterized up to equivalence by this property.

Moreover, we have an equivalence of the ambient $\left(\infty ,1\right)$-topos $H$ with the $\left(\infty ,1\right)$-category of etale geometric morphisms into it.

$\left(\left(\infty ,1\right)\mathrm{Topos}/H{\right)}_{\mathrm{et}}\simeq H\phantom{\rule{thinmathspace}{0ex}}.$((\infty,1)Topos/\mathbf{H})_{et} \simeq \mathbf{H} \,.
###### Example

Let $H=$ FinSet be the ordinary topos of finite sets. Then for $X\in \mathrm{FinSet}$ a finite set, a function object on $X$ is a morphism $\psi :\Psi \to X$ of sets. Under the cardinality decategorification

$\mid -\mid :\mathrm{FinSet}\to ℕ$|-| : FinSet \to \mathbb{N}

we think of this as the function

$\mid \psi \mid :X\to ℕ$|\psi| : X \to \mathbb{N}

given by

$x↦\mid {\Psi }_{x}\mid \phantom{\rule{thinmathspace}{0ex}},$x \mapsto |\Psi_x| \,,

where ${\Psi }_{x}\in \mathrm{FinSet}$ is the fiber of $\psi$ over $X$.

###### Example

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

$\infty \mathrm{Grpd}/X\simeq {\mathrm{PSh}}_{\left(\infty ,1\right)}\left(X\right)\simeq {\mathrm{Func}}_{\left(\infty ,1\right)}\left(X,\infty \mathrm{Grpd}\right)$\infty Grpd/X \simeq PSh_{(\infty,1)}(X) \simeq Func_{(\infty,1)}(X,\infty Grpd)

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.

## Products of function objects

For $\psi :\Psi \to X$ and $\varphi :\Phi \to X$ in $H/X$ two function objects on $X$, their product $\psi ×\varphi$ in $H/X$ we call the product of function objects.

This is computed in $H$ as the fiber product

$\psi {×}^{H/X}\varphi =\Psi {×}_{X}^{H}\Phi$\psi \times^{\mathbf{H}/X} \phi = \Psi \times^{\mathbf{H}}_X \Phi

and the morphism down to $X$ is the evident projection

$\begin{array}{ccc}& & \Psi {×}_{X}^{H}\Phi \\ & ↙& & ↘\\ \Psi & & {↓}^{\psi {×}^{H/X}\varphi }& & \Phi \\ & {}_{\psi }↘& & {↙}_{\varphi }\\ & & X\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ && \Psi \times_{X}^{\mathbf{H}} \Phi \\ & \swarrow && \searrow \\ \Psi &&\downarrow^{\psi \times^{\mathbf{H}/X} \phi}&& \Phi \\ & {}_{\mathllap{\psi}}\searrow && \swarrow_{\mathrlap{\phi}} \\ && X } \,.
###### Example

In $H=$ FinSet we have that the $ℕ$-valued function underlying the product function object is the usual pointwise product of functions

$\mid \psi \cdot \varphi \mid :x↦\mid \psi \mid \left(x\right)\cdot \mid \varphi \mid \left(x\right)\phantom{\rule{thinmathspace}{0ex}}.$|\psi \cdot \phi| : x \mapsto |\psi|(x) \cdot |\phi|(x) \,.

## Fiber integration

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

$\left({v}_{!}⊣{v}^{*}⊣{v}_{*}\right):H/X\stackrel{\stackrel{{v}_{!}}{\to }}{\stackrel{\stackrel{{v}^{*}}{←}}{\underset{{v}_{*}}{\to }}}H/Y$(v_! \dashv v^* \dashv v_*) : \mathbf{H}/X \stackrel{\overset{v_!}{\to}}{\stackrel{\overset{v^*}{\leftarrow}}{\underset{v_*}{\to}}} \mathbf{H}/Y

between the corresponding over-(∞,1)-toposes. Here ${v}_{!}$ acts simply by postcomposition with $v$:

${v}_{!}:\left(\Psi \stackrel{\psi }{\to }X\right)↦\left(\Psi \stackrel{\psi }{\to }X\stackrel{v}{\to }Y\right)$v_! : (\Psi \stackrel{\psi}{\to} X) \mapsto (\Psi \stackrel{\psi}{\to} X \stackrel{v}{\to} Y)

while ${v}^{*}$ acts by (∞,1)-pullback along $v$:

${v}^{*}:\left(\Phi \stackrel{\varphi }{\to }Y\right)↦\left(X{×}_{Y}\Phi \right)\phantom{\rule{thinmathspace}{0ex}}.$v^* : (\Phi \stackrel{\phi}{\to} Y) \mapsto (X \times_Y \Phi) \,.

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 adjoints and hence both preserve (∞,1)-colimits. Therefore these are morphism in Pr(∞,1)Cat and hence behave like linear maps on our function spaces $H/X$ and $H/Y$.

When we think of base change in the context of linear algebra on sheaves, we shall write ${\int }_{X/Y}:={v}_{!}$

$\left({\int }_{X/Y}⊣{v}^{*}\right):H/X\stackrel{\stackrel{{\int }_{X/Y}}{\to }}{\underset{{v}^{*}}{←}}H/Y$(\int_{X/Y} \dashv v^*) : \mathbf{H}/X \stackrel{\overset{\int_{X/Y}}{\to}}{\underset{v^*}{\leftarrow}} \mathbf{H}/Y

and call ${\int }_{X/Y}\psi$ the fiber integration of $F$ over the fibers of $v$. In particular when $Y=*$ is the terminal object we write simply

${\int }_{X}\psi \in H$\int_X \psi \in \mathbf{H}

for the integral of $\psi$ with values in the ambient $\left(\infty ,1\right)$-topos. (See also the notation for Lawvere distributions).

###### Example

Consider the ordinary topos $H=$ FinSet and for $X\in H$ any set the unique morphism $v:X\to *$ to the terminal object.

For $\psi :\Psi \to X$ a function object with underlying function $\psi :x↦\mid {\Psi }_{x}\mid$ we have that the integral

${\int }_{X}\psi :\Psi \to *$\int_X \psi : \Psi \to *

has as underlying function the constant

$\mid {\int }_{X}\psi \mid =\sum _{x\in X}\mid \psi \mid \left(x\right)\phantom{\rule{thinmathspace}{0ex}}.$|\int_X \psi| = \sum_{x \in X} |\psi|(x) \,.

## Integral transforms

If we are given an oriented span or correspondence

$\left(\begin{array}{ccc}& & A\\ & {}^{i}↙& & {↘}^{o}\\ X& & & & Y\end{array}\right)$\left( \array{ && A \\ & {}^{\mathllap{i}}\swarrow && \searrow^{\mathrlap{o}} \\ X &&&& Y } \right)

in $H$ it induces by composition of pullback and fiber integration operations a colimit-preserving $\left(\infty ,1\right)$-functor

$\underline{A}:H/X\stackrel{{i}^{*}}{\to }H/A\stackrel{{\int }_{A/Y}}{\to }H/Y\phantom{\rule{thinmathspace}{0ex}}.$\underline{A} : \mathbf{H}/X \stackrel{i^*}{\to} \mathbf{H}/A \stackrel{\int_{A/Y}}{\to} \mathbf{H}/Y \,.

We may always factor $\left(i,o\right)$ through the (∞,1)-product

$\left(\begin{array}{ccc}& & A\\ & & {↓}^{\left(i,o\right)}\\ & & X×Y\\ & {}^{{p}_{1}}↙& & {↘}^{{p}_{2}}\\ X& & & & Y\end{array}\right)\phantom{\rule{thinmathspace}{0ex}}.$\left( \array{ && A \\ && \downarrow^{\mathrlap{(i,o)}} \\ && X \times Y \\ & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}} \\ X &&&& Y } \right) \,.

We call the function object

$\left(\left(i,o\right):A\to X×Y\right)\in H/\left(X×Y\right)$((i,o) : A \to X \times Y) \in \mathbf{H}/(X\times Y)

on $X×Y$ the integral kernel of $\underline{A}$.

###### Observation

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

$\underline{A}F={\int }_{A/Y}{i}^{*}F=\left({p}_{2}{\right)}_{!}\left(A×\left({p}_{1}^{*}F\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\underline{A} F = \int_{A/Y} i^* F = (p_2)_!(A \times (p_1^* F) ) \,.
###### Proof

This follows from the pasting law for pullbacks in $H$:

$\begin{array}{ccccc}{i}^{*}\Psi & \to & {p}_{1}^{*}\Psi & \to & \Psi \\ ↓& & ↓& & {↓}^{\psi }\\ A& \stackrel{\left(i,o\right)}{\to }& X×Y& \stackrel{{p}_{1}}{\to }& X\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ 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 } \,.
###### Remark

By the above remark on etale geometric morphisms we have that we can recover the span $X\stackrel{i}{←}A\stackrel{o}{\to }Y$ in $H$ from the span

$\begin{array}{ccccc}H/X& \stackrel{\stackrel{{i}^{*}}{\to }}{\underset{{i}_{*}}{←}}& H/A& \stackrel{\stackrel{{o}^{*}}{←}}{\underset{{o}_{*}}{\to }}& H/Y\\ & ↘↖& ↓↑& ↙↗\\ & & H\end{array}$\array{ \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} }

in $\left(\left(\infty ,1\right)\mathrm{Topos}/H{\right)}_{\mathrm{et}}$.

###### Example

In $H=$ FinSet we have that $\left(\mid {A}_{x,y}\mid \right)$ is a $\mid X\mid$-by-$\mid Y\mid$-matrix with entries in natural numbers and the function

$\mid A\psi \mid :y↦\mid \left({i}^{*}\Psi {\right)}_{y}\mid =\sum _{x\in X}\mid {A}_{x,y}\mid \cdot \mid \psi \mid \left(x\right)$|A \psi | : y \mapsto | (i^* \Psi)_y | = \sum_{x \in X} |A_{x,y}| \cdot |\psi|(x)

is the result of applying the familiar linear map given by usual matrix calculus on $\mid \psi \mid$.

###### Example

In the case $H=$ ∞Grpd we have – as in the above example – by the (∞,1)-Grothendieck construction an equivalence

$\infty \mathrm{Grpd}/\left(X×Y\right)\simeq {\mathrm{PSh}}_{\left(\infty ,1\right)}\left(X×Y\right)\phantom{\rule{thinmathspace}{0ex}}.$\infty Grpd / (X \times Y) \simeq PSh_{(\infty,1)}(X \times Y) \,.

Since the $\infty$-groupoid $Y$ is equivalent to its opposite (∞,1)-category this may also be written as

$\infty \mathrm{Grpd}/\left(X×Y\right)\simeq {\mathrm{Func}}_{\left(\infty ,1\right)}\left(X×{Y}^{\mathrm{op}},\infty \mathrm{Grpd}\right)\phantom{\rule{thinmathspace}{0ex}}.$\infty Grpd / (X \times Y) \simeq Func_{(\infty,1)}(X \times Y^{op}, \infty Grpd) \,.

The objects on the right we may again think of as $\left(\infty ,1\right)$-profunctors $X⇸Y$. So in particular the kernel $\left(A\to X×Y\right)\in \infty \mathrm{Grpd}/\left(X×Y\right)$ is under this equivalence on the right hand identified with an $\left(\infty ,1\right)$-profunctor

$\stackrel{˜}{A}:X⇸Y\phantom{\rule{thinmathspace}{0ex}}.$\tilde A : X &#x21F8; Y \,.
Revised on May 30, 2011 22:25:00 by Urs Schreiber (131.211.239.155)