(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
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.
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.
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.
For $C$ a small (∞,1)-category the (∞,1)-category of (∞,1)-presheaves
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.
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.
For $C, D \in$ Pr(∞,1)Cat we have that $Func^L(C,D)$ is itself locally presentable.
See Pr(∞,1)Cat for details.
This means that to the extent that we may think of $C, D$ as analogous to vector spaces, also the space of linear maps between them is analogous to a vector space.
For $C$ and $D$ two locally presentable (∞,1)-categories there is locally presentable $(\infty,1)$-category $C \otimes D$ and an (∞,1)-functor
which is universal with respect to the property that it preserves (∞,1)-colimits in both arguments.
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.
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 $X$
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.
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
we think of this as the function
given by
where $\Psi_x \in FinSet$ is the fiber of $\psi$ over $X$.
Let $\mathbf{H} =$ ∞Grpd. By the (∞,1)-Grothendieck construction we have for $X \in \infty Grpd$ an ∞-groupoid an equivalence of (∞,1)-categories
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.
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
and the morphism down to $X$ is the evident projection
In $\mathbf{H} =$ FinSet we have that the $\mathbb{N}$-valued function underlying the product function object is the usual pointwise product of functions
For every morphism $v : X \to Y$ in the ambient (∞,1)-topos $\mathbf{H}$ there is the corresponding base change geometric morphism
between the corresponding over-(∞,1)-toposes. Here $v_!$ acts simply by postcomposition with $v$:
while $v^*$ acts by (∞,1)-pullback along $v$:
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 $\int_{X/Y} := v_!$
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
for the integral of $\psi$ with values in the ambient $(\infty,1)$-topos. (See also the notation for Lawvere distributions).
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
has as underlying function the constant
If we are given an oriented span or correspondence
in $\mathbf{H}$ it induces by composition of pullback and fiber integration operations a colimit-preserving $(\infty,1)$-functor
We may always factor $(i,o)$ through the (∞,1)-product
We call the function object
on $X \times Y$ the integral kernel of $\underline{A}$.
We have the pull-tensor-push formula for $\underline{A}$:
This follows from the pasting law for pullbacks in $\mathbf{H}$:
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
in $((\infty,1)Topos/\mathbf{H})_{et}$.
In $\mathbf{H} =$ FinSet we have that $(|A_{x,y}|)$ is a $|X|$-by-$|Y|$-matrix with entries in natural numbers and the function
is the result of applying the familiar linear map given by usual matrix calculus on $|\psi|$.
In the case $\mathbf{H} =$ ∞Grpd we have – as in the above example – by the (∞,1)-Grothendieck construction an equivalence
Since the $\infty$-groupoid $Y$ is equivalent to its opposite (∞,1)-category this may also be written as
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