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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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  • $\begingroup$ 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. $\endgroup$ Commented Jan 10, 2012 at 9:35
  • $\begingroup$ 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. $\endgroup$ Commented Jan 10, 2012 at 10:32
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    $\begingroup$ I like your question. But I'll let others more expert than me answer. $\endgroup$
    – David Roberts
    Commented Jan 10, 2012 at 11:18
  • $\begingroup$ @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. $\endgroup$ Commented Jan 10, 2012 at 11:53
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    $\begingroup$ @Martin: For the moment, the analogy is just formal to me. For the topos that I write $\mathbb{A}$ Mac Lane and Moerdijk write $S[U]$ in analogy to a polynomial ring because the set of morphisms from the polynomial algebra over a ground ring to another algebra is just the set of elements of that other algebra as the category of morphisms from a topos to $S[U]$ is the category of objects of that topos. I doesn't like that notation too much as one has to turn arrows around so that one should introduce, at least formally, somewhere a $\operatorname{Spec}$. $\endgroup$ Commented Jan 10, 2012 at 21:44

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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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