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The category of finite sets is not a Grothendieck topos, but its Ind category Ind(Finite-Sets) = Sets is a Grothendieck topos. Similarly, given a pro-finite group G, the Grothendieck topos of discrete G-sets is the Ind of the category of finite G-sets.

Question: Are there simple conditions on a category C which guaranty that Ind(C) is a Grothendieck topos?

The examples which interests me is of the following kind: I have an infinite sequence of finite groups $(G_i)$ and a corresponding sequence of quotient groups $G_i \to H_i$. I consider the category of sequences $(X_i)$ of finite sets such that each $X_i$ carries a $G_i$-action which factors through $H_i$ for almost all $i$. Even for extreme cases (i.e. $H_i=G_i$, $H_i=1$, or even $G_i=H_1=1$) I don't know if the corresponding Ind-category is a topos. Another possible variant is to require a uniform bound on the size of the $X_i$'s.

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I believe every coherent topos is the Ind completion of its subcategory of coherent objects. This covers classifying toposes of profinite groups and other similar examples. – Benjamin Steinberg Jun 19 '12 at 13:11
Thank you for your answer, Is there a way to identify a category as a category of coherent objects? Is there a good reference on the subject of coherent objects/topoi ? – Tomer Schlank Jun 19 '12 at 17:56
The standard books like Jounstone or Mac Lane Moerdijk discuss coherent toposes and objects. I am not sure if they axiomatize them. Topos theory is not my specialty. – Benjamin Steinberg Jun 19 '12 at 18:27

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