This is a refined/sheafified version of this [previos question](https://mathoverflow.net/questions/67121/sections-of-2-vector-bundles) of mine.


Let $(X,\mathcal{O}\_X)$ be a ringed space or more in general a ringed stack, where the structure sheaf $\mathcal{O}\_X$ is a sheaf of $\mathbb{K}$-algebras for some field $\mathbb{K}$. Then a global section of $\mathcal{O}\_X$ can be thought of as a "scalar field" on $X$. In the "categorical progression" of higher $\mathbb{K}$-vector spaces, the field $\mathbb{K}$ is the 0th level. 1st level are classical $\mathbb{K}$-vector spaces, and so the "1st level version" of a section of $\mathcal{O}\_X$ is (at least roughtly) a field of $\mathbb{K}$-vector spaces on $X$. A natural formalization of this rough idea is that of going from the sheaf $\mathcal{O}\_X$ to the stack $\mathcal{O}\_X$-Mod of sheaves of $\mathcal{O}\_X$-modules on $X$ (maybe with suitable assumptions, e.g., one could be considering only quasi-coherent sheaves of $\mathcal{O}\_X$-modules). A remarkable property of $\mathcal{O}\_X$-modules is that they can be pushed forward along morphisms of ringed spaces. Now, the structure sheaf $\mathcal{O}\_X$ we started with is in particular an $\mathcal{O}\_X$-module, and pushing it forward along the terminal morphism $\pi:X\to \{pt\}$ we precisely get sections of $\mathcal{O}\_X$ mentioned above. 


Now we can make a further step, and go from the stack of $\mathcal{O}\_X$-modules to the 2-stack of $\mathcal{O}\_X$-algebras (with $\mathcal{O}\_X$-$\mathcal{O}\_X$-bimodules as 1-morphisms and morphisms of bimodules as 2-morphisms). My question is: can $\mathcal{O}\_X$-algebras be pushed forward along morphisms of ringed spaces? (under which hypothesis?). In particular, considering $\mathcal{O}\_X$ as an $\mathcal{O}\_X$-algebra, what is the $\mathbb{K}$-algebra $\pi\_\*\mathcal{O}\_X$? (the prototypical conjectural example of this in my mind is the following: if $G$ is a finite group, then $\pi\_\*\mathcal{O}\_{pt//G}$ is Morita equivalent to $\mathbb{K}[G]$, the group algebra of $G$).


In the above, an $\mathcal{O}\_X$-algebra is to be thought as of a placeholder for its category of modules, so in a non-affine situation it is actually deceitful to reason in terms of algebras and a better description would be thinking of the "second step" as $\mathcal{O}\_X$-linear categories, with additive functors and natural transformations of these. In this more general setting, the pushforward to a point of $\mathcal{O}\_X$ would be the "pushforward of the 2-category of $\mathcal{O}\_X$-linear categories" and this should be a $\mathbb{K}$-linear category, but not necessarily the category of representations of an algebra. In particular, by abstract nonsense I would expect this pushforward to be the category of $\mathcal{O}\_X$-modules.


Note that going one step backwards instead of one step forward, the existence of a pushforward is nontrivial: the pushforward of a section of $\mathcal{O}\_X$ along $X\to \{pt\}$ is an integration of fuctions on $X$, so it requires additional data to be defined (a "measure").



Pushforwards of stacks of algebras?
Yes, there are several different formalisms that achieve this - for example it's treated in Lurie's DAG XI or Toen-Vezzosi [Caractères de Chern, traces équivariantes et géométrie algébrique dérivée](http://arxiv.org/abs/0903.3292) in the derived context, and in many places (eg Gaitsgory [The notion of category over an algebraic stack](http://arxiv.org/abs/math/0507192)) in the underived context. Basically as I understand your question you're looking at an appropriate class of quasicoherent sheaves of categories on a variety or stack $X$ - which, given enough compact objects (see eg Toen's derived Azumaya paper) are monadic over $QC(X)$, i.e. are the same as module categories for $O\_X$-algebras.


But in fact such objects can be said much more concretely: all geometric stacks
(eg quasicompact with affine diagonal) are "affine" from the point of view of these sheaves, ie there's no difference between such a quasicoherent sheaf of categories and its global sections (this is in DAG XI).


So one can just take the following as definition: we're just looking at module categories for the symmetric monoidal (dg or $\infty$- if you like) category $QC(X)$ of quasicoherent sheaves on $X$. In this language pushforward and pullback of the kind you ask are super easy, just as they are for modules over ordinary rings: given a map $X\to Y$ you have a "homomorphism" (symmetric monoidal functor) $QC(Y)\to QC(X)$ (pullback), and you can use this to forget $QC(X)$ modules down to $QC(Y)$, i.e. push forward, or (the left adjoint of this) tensor a $QC(Y)$-module by $QC(X)$ (pullback).


Moreover under some hypotheses ($X\to Y$ faithfully flat or $X\to Y$ proper and $X,Y$ smooth) descent holds - i.e. quasicoherent sheaves of categories on $Y$ are the same as those on $X$ with descent data. Said another way there's a Morita equivalence of monoidal categories between $QC(Y)$ and $QC(X\times\_Y X)$ (theorem of Francis, Nadler and myself
which will be posted, at least on my webpage, sometime in the next couple of weeks).
In the case of $pt \to pt/G$ this recovers a Morita equivalence between categories with$G$ action and categories over $BG$.


