nLab integral transforms on sheaves



Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos



There is a sense in which a sheaf FF is like a categorification of a function: We may think of the stalk-map from topos points to sets (x *x *)F(x):=x *FSet(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.

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


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

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

is the free (∞,1)-cocompletion of CC, 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 CC.

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


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

See profunctor for details.



For C,DC, D \in Pr(∞,1)Cat we have that Func L(C,D)Func^L(C,D) is itself locally presentable.

See Pr(∞,1)Cat for details.


This means that to the extent that we may think of C,DC, D as analogous to vector spaces, also the space of linear maps between them is analogous to a vector space.

Tensor products


For CC and DD two locally presentable (∞,1)-categories there is locally presentable (,1)(\infty,1)-category CDC \otimes D and an (∞,1)-functor

C×DCD C \times D \to C \otimes D

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


This means that in as far as C,DC, D \in Pr(∞,1)Cat are analogous to vector spaces, CDC \otimes D is analogous to their tensor product.

Function spaces

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

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


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

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

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

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

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

we think of this as the function

|ψ|:X |\psi| : X \to \mathbb{N}

given by

x|Ψ x|, x \mapsto |\Psi_x| \,,

where Ψ xFinSet\Psi_x \in FinSet is the fiber of ψ\psi over XX.


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

Grpd/XPSh (,1)(X)Func (,1)(X,Grpd) \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 XX with the (∞,1)-category of (∞,1)-presheaves on XX. And since the \infty-groupoid CC is equivalent to its opposite (∞,1)-category this is also equivalent to the (∞,1)-category of (∞,1)-functors from CC to ∞Grpd.

Products of function objects

For ψ:ΨX\psi : \Psi \to X and ϕ:ΦX\phi : \Phi \to X in H/X\mathbf{H}/X two function objects on XX, their product ψ×ϕ\psi \times \phi in H/X\mathbf{H}/X we call the product of function objects.

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

ψ× H/Xϕ=Ψ× X HΦ \psi \times^{\mathbf{H}/X} \phi = \Psi \times^{\mathbf{H}}_X \Phi

and the morphism down to XX is the evident projection

Ψ× X HΦ Ψ ψ× H/Xϕ Φ ψ ϕ X. \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 } \,.

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

|ψϕ|:x|ψ|(x)|ϕ|(x). |\psi \cdot \phi| : x \mapsto |\psi|(x) \cdot |\phi|(x) \,.

Fiber integration

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

(v !v *v *):H/Xv *v *v !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 !v_! acts simply by postcomposition with vv:

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

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

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

There is a further right adjoint v *v_*. For the present purpose the relevance of its existence is that it implies that both v !v_! as well as v *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\mathbf{H}/X and H/Y\mathbf{H}/Y.

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

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

and call X/Yψ\int_{X/Y} \psi the fiber integration of FF over the fibers of vv. In particular when Y=*Y = * is the terminal object we write simply

XψH \int_X \psi \in \mathbf{H}

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


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

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

Xψ:Ψ* \int_X \psi : \Psi \to *

has as underlying function the constant

| Xψ|= xX|ψ|(x). |\int_X \psi| = \sum_{x \in X} |\psi|(x) \,.

Integral transforms

If we are given an oriented span or correspondence

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

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

A̲:H/Xi *H/A A/YH/Y. \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)(i,o) through the (∞,1)-product

( A (i,o) X×Y p 1 p 2 X Y). \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):AX×Y)H/(X×Y) ((i,o) : A \to X \times Y) \in \mathbf{H}/(X\times Y)

on X×YX \times Y the integral kernel of A̲\underline{A}.


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

A̲F= A/Yi *F=(p 2) !(A×(p 1 *F)). \underline{A} F = \int_{A/Y} i^* F = (p_2)_!(A \times (p_1^* F) ) \,.

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

i *Ψ p 1 *Ψ Ψ ψ A (i,o) X×Y p 1 X. \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 } \,.

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

H/X i *i * H/A o *o * H/Y H \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 ((,1)Topos/H) et((\infty,1)Topos/\mathbf{H})_{et}.


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

|Aψ|:y|(i *Ψ) y|= xX|A x,y||ψ|(x) |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 |ψ||\psi|.


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

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

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

Grpd/(X×Y)Func (,1)(X×Y op,Grpd). \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 (,1)(\infty,1)-profunctors XYX ⇸ Y. So in particular the kernel (AX×Y)Grpd/(X×Y)(A \to X \times Y) \in \infty Grpd/(X \times Y) is under this equivalence on the right hand identified with an (,1)(\infty,1)-profunctor

A˜:XY. \tilde A : X ⇸ Y \,.

Last revised on July 14, 2019 at 07:48:34. See the history of this page for a list of all contributions to it.