Is there an $(\infty,2)$-category with morphisms given by $D^b\text{Coh}$?

Question

My question is:

Has anyone constructed an $(\infty,2)$-category whose objects are (projective, maybe smooth, ...) varieties, and where the 1-morphisms from $X$ to $Y$ are given by $D^b_\infty\text{Coh}(X \times Y)$ (an $(\infty,1)$-enhancement of $D^b\text{Coh}(X \times Y)$)?

By "$(\infty,1)$-enhancement" I mean some $(\infty,1)$-category whose homotopy category is $D^b\text{Coh}$.

I would hope that binary composition would descend to the functor $$D^b\text{Coh}(Y\times Z) \times D^b\text{Coh}(X \times Y) \to D^b\text{Coh}(X \times Z)$$ that sends $(Q,P)$ to $\pi_{02,*}(\pi_{01}^*P \otimes \pi_{12}^*Q)$ a la the Fourier--Mukai transform (where tensor and the projections are the derived functors).

If the answer is "yes", I wonder whether this is special to derived categories of coherent sheaves or whether it's a more general algebraic result about collections of $(\infty,1)$-categories of a certain kind?

I'm new to infinity infinity stuff and I might have misused a technical term, so it might be best not to interpret my words too literally.

Answer 1

For smooth projective varieties it is known that $$D^b_\infty Coh(X\times Y)\simeq Hom(D^b_\infty Coh(X), D^b_\infty Coh(Y))$$ compatibly with the composition you describe (note here $D^b_\infty Coh$ coincides with the $\infty$-category $Perf$ of perfect complexes). Hence your desired $(\infty,2)$-category is a full subcategory of the $(\infty,2)$-category of dg categories (or small idempotent-complete stable $\infty$-categories) where the above Homs take place. The analogous result for $QCoh$ holds in much greater generality (without smoothness or projectivity - eg for quasicompact quasiseparated schemes).

Edit: Without smoothness, you find (here) that $D^b_\infty Coh(X\times Y)$ represents functors from $Perf(X)$ to $D^b_\infty Coh(Y)$, and that functors from $D^b_\infty Coh(X)$ to $D^b_\infty Coh(Y)$ are represented by kernels which are coherent relative to the first factor. Note you have to be careful in the formulation of the question since $D^b_\infty Coh$ is not preserved by tensor product on a singular variety! (e.g. self-tor of a skyscraper at a singular point is not bounded, i.e., not in $D^b_\infty Coh$)