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This is a refined/sheafified version of this previos question 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").

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Given a morphism of ringed spaces, the pushforward of module sheaves is lax monoidal and therefore restricts to the category of algebras. See for example EGA I 4.2. But I assume that you know that and I've misunderstood the question ... –  Martin Brandenburg Apr 17 '12 at 17:50
    
Hi Martin, thanks. Here I'm thinking of $\mathcal{O}_X$-algebras as the 2-category having $\mathcal{O}_X$-$\mathcal{O}_X$-bimodules as 1-morphisms and bimodule morphisms as 2-morphisms, not of 𝕆X-algebras as a subcategory of $\mathcal{O}_X$-modules. But I may be misunderstanding your comment: let me know about this. –  domenico fiorenza Apr 17 '12 at 18:02
    
Ok, but the objects are just algebras? But doesn't the usual pushforward extend to your $2$-category? –  Martin Brandenburg Apr 17 '12 at 18:10
    
Yes, but in this context an algebra is to be thought as a placeholder for its category of modules. Now that I write it I see that in the non-affine case one should not expect an algebra just like a quasi-coherent $\mathcal{O}_X$-module on $X$ is not the sheaf of modules associated with an $\mathcal{O}_X(X)$-module. I'll now edit accordingly my question. Thanks again. –  domenico fiorenza Apr 17 '12 at 19:21
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up vote 5 down vote accepted

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 in the derived context, and in many places (eg Gaitsgory The notion of category over an algebraic stack) 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 $BG$.

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Thanks a lot, David! –  domenico fiorenza Apr 18 '12 at 5:39
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