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I am wondering if prestacks over a fixed category, or stacks over a fixed site, form an (interesting) category, equipped with prestacks morphisms. First of all, do they form a proper class or a set?

Also, since sheaf categories on a topological space form a topos, I would like to know if there is a similar (and similarly powerful) notion for stacks, which in many senses are parallel to sheaves. So: do stacks over a fixed site form a topos? And if not, do they form an interesting category?

My question is formulated for topological stacks. Anyway, also answers in the context of algebraic stacks are welcome.

Thank you in advance. Sorry if the question is too vague, but my tools do not allow me to formulate more precise statements.

W.Rether
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    Stacks in groupoids over an arbitrary small site do not form a topos, but they do form a (2,1)-topos, or, if you prefer, an ∞-topos. See Lurie's Higher Topos Theory for additional information. – Dmitri P. Sep 02 '18 at 03:23
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    Dmitri is right. As an example, the $\infty$-category of stacks-in-groupoids over the one-point space is the $\infty$-category of groupoids, the prototypical example of an $\infty$-topos (just as the category of sets is the prototypical example of a $1$-topos). – Ingo Blechschmidt Sep 02 '18 at 10:43
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    It seems that both comments above are conflating groupoids with $\infty$-groupoids. The stacks of groupoids over a point surely form the $\infty$-category of groupoids, which is only a $2-$category and a $2$-topis. IIRC, it's impossible for a lower topos to form an $\infty$-topos, as not all colimits can be van Kampen. More concretely, the localization functor $\Pi_1$ from $\infty$-groupoids to groupoids doesn't preserve finite limits, such as the loop space of any space with nontrivial $\pi_2$. – Kevin Carlson Sep 03 '18 at 20:23

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