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.

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.

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.

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.

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 ->> 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.

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.

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.