Topos associated to a category

For each topos $\mathbb E$ let $\mathcal O(\mathbb E)$ be the locally presentable category of objects in $\mathbb E$. We can make $\mathcal O$ into a contravariant functor to the category of locally presentable categories (with morphisms being cocontinuous) by assigning to each geometric morphism $(f^*, f_*)$ the functor $f^*$. By [Mac Lane, Moerdijk: Sheaves in Geometry and Logic] this functor is representable, that is there is a topos $\mathbb A$, called the object classifier, such that there is a natural equivalence $$\mathrm{Hom}(\mathbb E, \mathbb A) \to \mathcal O(\mathbb E).$$ Now I wonder whether $\mathcal O$ has a right adjoint, which I want to call $\operatorname{Spec}$ due to the analogy with algebraic geometry, that is whether there exists a contravariant functor $\operatorname{Spec}$ from the category of locally presentable categories to the category of topoi (with geometric morphisms) such that there is a natural equivalence $$\mathrm{Hom}(\mathbb E, \operatorname{Spec}\mathcal C) \to \mathrm{Hom}(\mathcal C, \mathcal O(\mathbb E))$$ of categories.

(Here, topos shall mean Grothendieck topos.)

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I guess you want to cut down to small categories C, in order to get a Spec(C) satisfying your requirements. When C is small, the theory of diagrams on C is a geometric theory, and therefore has a classifying topos Spec(C). So (modulo the size question), I think the answer to your question must be yes. But I'll let others more expert than me answer. – Tom Leinster Jan 10 2012 at 9:35
You should be willing to change one of the $\mathrm{Hom}$ to something else, such as "continuous functors". But I'll let others more expert than me answer. – Andrej Bauer Jan 10 2012 at 10:32
I like your question. But I'll let others more expert than me answer. – David Roberts Jan 10 2012 at 11:18
@Tom: You are right; in my question I am a bit sloppy when it comes to size issues. @Andrej: Do you possibly mean cocontinuous? I will changed my question to address both comments in a manner that is hopefully helpful. – Marc Nieper-Wißkirchen Jan 10 2012 at 11:53
There is some hesitation between terminology continuous and cocontinuous. Bass used right and left continuous and right continuous was more important in module theory, so some retained continuous for what pure category theorists say cocontinuous. See also Lurie's book which also has it that way (and also Rosenberg who follows Bass). – Zoran Škoda Jan 10 2012 at 12:04

This is described in the paper

Bunge and Carboni, The symmetric topos, Journal of Pure and Applied Algebra 105:233-249, 1995.

which describes the sense in which the construction you call Spec is analogous to the symmetric algebra construction.

Bunge and Carboni give a biadjunction between the bicategory R, which is the opposite of the bicategory of Grothendieck toposes, and the bicategory A of locally presentable categories and cocontinuous functors (equivalently, left adjoints).

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 Thanks! That's exactly the kind of answer I hoped for! – Marc Nieper-Wißkirchen Jan 11 2012 at 9:10