12
$\begingroup$

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.

Improve this question
$\endgroup$
3
  • 2
    $\begingroup$ 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. $\endgroup$
    – Benjamin Steinberg
    Commented Jun 19, 2012 at 13:11
  • $\begingroup$ 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 ? $\endgroup$
    – Tomer Schlank
    Commented Jun 19, 2012 at 17:56
  • 1
    $\begingroup$ 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. $\endgroup$
    – Benjamin Steinberg
    Commented Jun 19, 2012 at 18:27

1 Answer 1

Reset to default
8
$\begingroup$

In Di Liberti–Ramos González's Gabriel-Ulmer duality for topoi and its relation with site presentations, they state (Theorem 3.17) that $\mathbf{Ind}(\mathscr C)$ is a Grothendieck topos if and only if $\mathscr C$ is extensive and pro-exact (Definition 3.14 therein).

They attribute this result to Carboni–Pedicchio–Rosický's Syntactic characterizations of various classes of locally presentable categories, though it does not appear as an explicit theorem there, so it not such a convenient reference.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .