Geometric function theory considers notions of higher generalized functions on higher generalized spaces (such as on groupoids, on orbifolds and more generally on infinity-stacks) such that all suitably generalized linear maps between the monoidal $\infty$-structures of functions on two spaces arise from a higher analog of plain matrix multiplication, namely from a pull-tensor-push operation in the given $\infty$-context.
The motivating toy example is that where all spaces in question are just finite sets and functions are just maps from finite sets to a given ground field: for $X$ a finite set the collection of functions $C(X)$ on $X$ is canonically identified with the vector space spanned by the elements of $X$. Moreover, for two finite sets $X_1$ and $X_2$ functions on the product set $X_1 \times X_2$ are canonically identified with $|X_1| \times |X_2|$-matrices
The point is that from this perspective on matrices the action of a matrix on a vector is given by the pull-tensor-push operation of functions through a span
Given $v \in C(X_1)$ and $M \in C(X_1 \times X_2)$ the matrix product $M \cdot v \in C(X_2)$ can be understood as
where
$pr_1^* : C(X_1) \to C(X_1 \times X_2)$ is the pullback of functions along the projection map $pr_1$;
$M \cdot (\cdot)$ is the ordinary product of functions;
$\int_{pr_2}$ is the push-forward of field-valued functions along $pr_2$, i.e. the operation which integrates (a finite sum in this toy example) a function on $X_1 \times X_2$ over $X_1$ to a function on $X_2$.
A closely related situation is considered in the context of groupoidification, where the aim is to encode not only the operation of matrix multiplication geometrically in the above sense, but also the very notion of a function itself.
In groupoidification functions with values in fields are essentially replaced by functions with values in (finite) groupoids, and the notion of groupoid cardinality is used to interpret such a gadget as a function with values in rational numbers.
More precisely, groupoidification in the sense of John Baez can be understood as geometric function theory for the case that collections of geometric functions are modeled as over categories. This is described in more detail at examples for geometric function objects.
Related toy examples of the general pattern are group algebras regarded as category algebras (see the end of that entry for the span-description of category algebras).
In full generality the idea of geometric function theory is to replace in the above toy example all finite sets with generalized higher spaces in the form of ∞-stacks and to realize a useful notion of higher “functions” $C(X)$, such that $C(X)$ is a monoidal (∞,1)-category in a suitable sense and behaves as expected under homotopy pullback and push-forward operations.
The main two conditions that one wants to have in geometric function theory are
for $X_1 \to Y \leftarrow X_2$ two generalized spaces sitting over a third one, we have an equivalence between the generalized functions on the fiber product $X_1 \times_Y X_2$ and the tensor product of functions on $X_1$ with functions on $X_2$ over functions on $Y$:
where all operations are in the suitable $\infty$-context (for instance the fiber product is a homotopy limit).
again for given $X_1 \to Y \leftarrow X_2$ the generalized functions on the (homotopy/$\infty$-)fiber product are supposed to induce via (homotopy/$\infty$-) pull-tensor-push operation through the span $X_1 \leftarrow X_1 \times_{Y} X_2 \to X_2$ all $C(Y)$-linear morphism between $C(X_1)$ and $C(X_2)$:
This is to be read as a vast generalization of ordinary matrix multiplication, to which it reduces in the case that all spaces here are finite sets and all functions are ordinary functions on finite sets with values in some field. One step higher this gives Fourier–Mukai transformations? etc.
A particularly interesting application of items 1) and 2) above is to the case where the pullback diagram of generalized spaces in question is simply
As discussed at span trace the homotopy pullback of this is the loop space object $\Lambda X$ of $X$.
So in this case statement 2) relates
with
The expression on the right is known as the higher trace of the monoidal (∞,1)-category $C(X)$, which is the higher Hochschild homology of $C(X)$. (See next subsection for concrete realizations).
Aspects of geometric function theory have a long history in the context of sheaves and stacks over algebraic sites. The Fourier–Mukai transform? is a classical example of a pull-tensor-push operation in a context where the object $C(X)$ of generalized functions on a space $X$ is taken to be $D(X)$, the derived category of coherent sheaves on $X$.
A general proof of the above item 2 in the context of generalized functions modeled by the derived category $D(X)$ has been given in
As any homotopy category, the derived category $D(X)$ of coherent sheaves is naturally understood as the $1$-categorical shadow of a stable (∞,1)-category. In a suitable context of derived $\infty$-stacks there is for each such $\infty$-stack $X$ an $(\infty,1)$-category $QC(X)$ whose homotopy category is the ordinary derived category $D(X)$ of quasi-coherent sheaves on $X$.
In this context geometric function theory with functions $C(X)$ taken to be given by $QC(X)$ was discussed in
where the above two items appear as items 1) and 2) in theorem 1.2, p. 4,5
A detailed entry on this is at
Further discussion of realizations of geometric function objects is at
Closely related to the spans appearing in geometric function theory is the notion of bi-brane.
David Ben-Zvi kindly wrote an expositional piece on geometric function theory for the $n$-Category Café (blog entry):