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 $D^b\text{Coh}(X \times Y)$)?
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> 
> 


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.



Is there an $(\infty,2)$-category with morphisms given by $D^b\text{Coh}$?
For smooth projective varieties [it is known](http://arxiv.org/abs/math/0408337)  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](http://arxiv.org/abs/1312.7164)) 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 $D^b\_\infty Coh$) 


