nLab
integral transforms on sheaves

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 topos points to sets $(x^* \dashv x_*) \mapsto F(x) := x^* F \in Set$ 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 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

Every locally presentable (∞,1)-category is a reflective sub-(∞,1)-category of an (∞,1)-category of (∞,1)-presheaves.

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

Remark

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

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

is the free (∞,1)-cocompletion of $C$, 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 $(\infty,1)$-category is analogous in this sense to a sub-space of a vector space spanned by a basis.

Proposition

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 $C ⇸ D$.

See profunctor for details.

Hom-spaces

Proposition

For $C, D \in$ Pr(∞,1)Cat we have that $Func^L(C,D)$ 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 $(\infty,1)$-category $C \otimes D$ and an (∞,1)-functor

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

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 $\mathbf{H}$.

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 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 $X$

Remark

The over-(∞,1)-toposes $\mathbf{H}/X$ 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 morphisms into it.

$$((\infty,1)Topos/\mathbf{H})_{et} \simeq \mathbf{H} \,.$$

Example

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

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

we think of this as the function

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

given by

$$x \mapsto |\Psi_x| \,,$$

where $\Psi_x \in FinSet$ is the fiber of $\psi$ over $X$.

Example

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

$$\infty Grpd/X \simeq PSh_{(\infty,1)}(X) \simeq Func_{(\infty,1)}(X,\infty Grpd)$$

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

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

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

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

$$\array{ && \Psi \times_{X}^{\mathbf{H}} \Phi \\ & \swarrow && \searrow \\ \Psi &&\downarrow^{\psi \times^{\mathbf{H}/X} \phi}&& \Phi \\ & {}_{\mathllap{\psi}}\searrow && \swarrow_{\mathrlap{\phi}} \\ && X } \,.$$

Example

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

$$|\psi \cdot \phi| : x \mapsto |\psi|(x) \cdot |\phi|(x) \,.$$

Fiber integration

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

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

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

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

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

$$v^* : (\Phi \stackrel{\phi}{\to} Y) \mapsto (X \times_Y \Phi) \,.$$

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 $\mathbf{H}/X$ and $\mathbf{H}/Y$.

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

$$(\int_{X/Y} \dashv v^*) : \mathbf{H}/X \stackrel{\overset{\int_{X/Y}}{\to}}{\underset{v^*}{\leftarrow}} \mathbf{H}/Y$$

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

$$\int_X \psi \in \mathbf{H}$$

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

Example

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

$$\int_X \psi : \Psi \to *$$

has as underlying function the constant

$$|\int_X \psi| = \sum_{x \in X} |\psi|(x) \,.$$

Integral transforms

If we are given an oriented span or correspondence

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

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

$$\underline{A} : \mathbf{H}/X \stackrel{i^*}{\to} \mathbf{H}/A \stackrel{\int_{A/Y}}{\to} \mathbf{H}/Y \,.$$

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

$$\left( \array{ && A \\ && \downarrow^{\mathrlap{(i,o)}} \\ && X \times Y \\ & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}} \\ X &&&& Y } \right) \,.$$

We call the function object

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

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

Observation

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

$$\underline{A} F = \int_{A/Y} i^* F = (p_2)_!(A \times (p_1^* F) ) \,.$$

Proof

This follows from the pasting law for pullbacks in $\mathbf{H}$:

$$\array{ i^* \Psi &\to& p_1^* \Psi &\to& \Psi \\ \downarrow && \downarrow && \downarrow^{\mathrlap{\psi}} \\ A &\stackrel{(i,o)}{\to}& X \times Y &\stackrel{p_1}{\to}& X } \,.$$

Remark

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

$$\array{ \mathbf{H}/X &\stackrel{\overset{i^*}{\to}}{\underset{i_*}{\leftarrow}}& \mathbf{H}/A &\stackrel{\overset{o^*}{\leftarrow}}{\underset{o_*}{\to}}& \mathbf{H}/Y \\ & \searrow\nwarrow & \downarrow\uparrow & \swarrow\nearrow \\ && \mathbf{H} }$$

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

Example

In $\mathbf{H} =$ FinSet we have that $(|A_{x,y}|)$ is a $|X|$-by-$|Y|$-matrix with entries in natural numbers and the function

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

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

Example

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

$$\infty Grpd / (X \times Y) \simeq PSh_{(\infty,1)}(X \times Y) \,.$$

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

$$\infty Grpd / (X \times Y) \simeq Func_{(\infty,1)}(X \times Y^{op}, \infty Grpd) \,.$$

The objects on the right we may again think of as $(\infty,1)$-profunctors $X ⇸ Y$. 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

$$\tilde A : X ⇸ Y \,.$$

Related concepts

• integral transform

  • Fourier-Mukai transform, Hecke transform

• polynomial functor

• pure motives, motives in physics, motivic quantization

• path integral as a pull-push transform