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Certain Galois toposes can be written as $\lim_{i \in I} \mathbf{PSh}(G_i)$ where $(G_i)_{i \in I}$ is an inverse system of discrete groups. (The limit is a strict limit in the 2-category of Grothendieck toposes and geometric morphisms.)

What are the objects and morphisms in this topos?

If all groups $G_i$ in the diagram are finite and the transition maps are surjective, then I believe the objects of the topos are sets $S$ together with a $G$-action, with the special property that the action morphism $G \to \mathrm{End}(S)$ factors through one of the groups $G_i$. The morphisms are the functions such that $\phi(g\cdot x) = g\cdot \phi(x)$ for all $g\in G$. Can you do something similar in general?

Edit: One of the standard references for this is "Continuous fibrations and inverse limits of toposes". The idea there is to replace the diagram of Grothendieck toposes by a corresponding diagram of Grothendieck sites, with a special kind of transition maps (called continuous fibrations). Then a "limit Grothendieck site" is constructed and the inverse limit topos is the topos of sheaves on this site. But I don't know how to follow this process in practice, even in the case described above where all groups are finite and the transition maps between the groups are surjective.

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  • $\begingroup$ I'm pretty sure you take the colimit in the category of cocomplete and finitely complete categories (and cocontinuous, lex functors) of the corresponding diagram of toposes and the inverse image parts of the geometric morphisms between them induced by the diagram of groups. Then the adjoint functor theorem gives right adjoints to all the functors in the cocone, giving geometric morphisms as maps in a cone over the original diagram of Grothendieck toposes. $\endgroup$ – David Roberts Aug 28 '20 at 16:22
  • $\begingroup$ Thank you! Do you know how to compute the colimit in this category? For example, is it different from taking the colimit in the 2-category of categories? $\endgroup$ – Jens Hemelaer Aug 28 '20 at 16:37
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    $\begingroup$ I am sorry, I meant B3.4. $\endgroup$ – Ivan Di Liberti Aug 28 '20 at 20:29
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    $\begingroup$ @JensHemelaer Your description of the classifying topos of a profinite group is not correct (the category you describe does not have infinite sums, for example). See SGA4, IV.2.7. Also, the inclusion Top ⊂ Cat preserves filtered limits, so objects/morphisms in such a limit of toposes are just compatible families of objects/morphisms. $\endgroup$ – Marc Hoyois Aug 28 '20 at 21:18
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    $\begingroup$ @JensHemelaer Yes, the morphisms in Top are the pushforward functors. And "limit" should be understood in the 2-categorical sense, so a compatible family of objects includes isomorphisms satisfying the cocycle condition. $\endgroup$ – Marc Hoyois Aug 29 '20 at 13:47
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The question was solved thanks to comments by Marc Hoyois.

The classifying topos associated to a pro-group is described by Grothendieck and Verdier in SGA4, Exposé IV, 2.7 (link). Grothendieck and Verdier assume that the transition maps $\pi_{ij} : G_j \to G_i$ are surjective.

The classifying topos of a pro-group $(G_i)_{i \in I}$ as defined there, agrees with the inverse limit topos $\varprojlim_{i \in I} \mathbf{PSh}(G_i)$, by Remark 2.7 in "Prodiscrete groups and Galois toposes".

As suggested by მამუკა ჯიბლაძე below, here the construction of Grothendieck and Verdier in more detail:

The objects of $\varprojlim_{i \in I} \mathbf{PSh}(G_i)$ are sets $E$, written as directed union $\bigcup_{i \in I} E_i$ where each $E_i$ comes with a $G_i$-action, and such that $E_i = \{ x \in E_j : gx = x,~\forall g \in \ker(G_j \to G_i)\}$ and the action of $G_i$ on $E_i$ is the one induced by the action of $G_j$ on $E_j$ (this last part is left implicit in SGA4).

If we write $E_j$ as a direct union of orbits, then $E_i$ consists of the orbits isomorphic to $G_j/H$ with $H$ containing $\ker(G_j \to G_i)$. In this way, we get another description of the objects of $\varprojlim_{i \in I} \mathbf{PSh}(G_i)$. They are isomorphic to direct sums of quotients $G_i/H$, where two quotients $G_i/H$ and $G_j/H'$ are the same if there is a $k \leq i,j$ and a subgroup $H'' \subseteq G_k$ such that $\pi_{ik}(H'') = H$ and $\pi_{jk}(H'')=H'$.

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    $\begingroup$ It would be convenient to add the description itself, it is not so difficult: in the profinite case these are just continuous $\left(\varprojlim_iG_i\right)$-sets. In the general case (assuming all homomorphisms $G_j\to G_i$ surjective), one has a set $E$ covered by subsets $E_i$ with $G_i$-actions such that $E_i$ is the fixed set of the action of $\ker(G_j\to G_i)$ on $E_j$ for each $i\to j$. $\endgroup$ – მამუკა ჯიბლაძე Aug 31 '20 at 14:45
  • $\begingroup$ @მამუკაჯიბლაძე Good idea, but I realize that at the moment I don't understand the description yet. Shouldn't there be an additional condition that says that the actions of the different groups $G_i$ are compatible with each other? Consider for example the pro-group coming from $\mathbb{Z}/4\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z}$. Then the total set $E$ has a $\mathbb{Z}/4\mathbb{Z}$-action and it has a subset $E' \subseteq E$ fixed by $2\mathbb{Z}/4\mathbb{Z}$. But then according to SGA4 you have another piece of data involving a $\mathbb{Z}/2\mathbb{Z}$-action on $E'$? $\endgroup$ – Jens Hemelaer Aug 31 '20 at 17:17
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    $\begingroup$ I would bet they mean the canonical action: if $G$ acts on $E$ and $N$ is a normal subgroup of $G$, this gives rise to an action of $G/N$ on $E^N$. It is this one they must have in mind there. $\endgroup$ – მამუკა ჯიბლაძე Aug 31 '20 at 17:22

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