Relation between topos and $\infty$-topos

Question

I'm currently reading The book of Jacob Lurie, 'Higher Topos Theory', and I'm a little confused by the relation between classical topos and $\infty$-topos :

to an $\infty$-topos I can attach the ordinary topos of its $0$-truncated objects. And to a classical topos I have several way to associate $\infty$-topos 'above' it.

Jacob Lurie (in his book, section 6.4) present this relation as similar to the relation between a classical topos and its locale of sub-terminal objects. In this situation, I know I can have plenty of topoi (a proper class) that are associated to the same locale, even if this locale is just a point. But I have no idea of what happens in the case of $\infty$-Topos : I have seen that in some cases there might be several non equivalent $\infty$-topos above a same ordinary topos, but I see them more like "different ways of doing homotopy theory in the internal logic of $X$ because some classical result of homotopy theory (like Whitehead's theorem) may fail in the internal logic" rather than completely different objects that just share a small property" (like the class of topoi whose locale of subterminal objects is reduced to a point is just the class of topos whose internal logic is two-valued)

So for example : Are there several (non equivalent) $\infty$-topoi, whose topos of $0$-truncatued objects is the topos of set ? If it's the case, can I have an example ? Are we able to 'classify' them ?

Answer 1

For any space $X$, there's an $\infty$-topos of spaces fibered over $X$. The underlying ordinary topos is the category of representations of the fundamental groupoid of $X$. So if $X$ is simply connected, this is just the category of sets. But the $\infty$-topoi are different for different values of $X$ (two spaces $X$ and $Y$ yield equivalent $\infty$-topoi if and only if $X$ and $Y$ are homotopy equivalent).

Answer 2

Jacob's answer to your "for example" question is a good one, but let me be so bold as to try to address the general question. I think there are two different issues in play: the fact that the site of a topos may not be truncated, and the fact that an $(\infty,1)$-topos may not be hypercomplete. The first is just as much the case for 1-toposes, while the second is a purely $\infty$-phenomenon, and this may be the source of some of your confusion.

Every 1-topos is a left exact localization of a category of presheaves on some 1-category. If this 1-category is in fact a (0,1)-category (a poset), then the 1-topos is localic and equivalent to the topos of sheaves on its locale of subterminal objects, and the 2-category of localic 1-toposes is equivalent to that of locales. But, as you note, many different 1-toposes can have equivalent locales of subterminal objects, because we can consider 1-toposes of sheaves on 1-categories that are not posets.

Similarly, every $(\infty,1)$-topos is a left exact localization of a category of presheaves on some $(\infty,1)$-category. If this $(\infty,1)$-category is in fact a 1-category and the localization is topological, then the $(\infty,1)$-topos is 1-localic and equivalent to the $(\infty,1)$-topos of sheaves on its 1-topos of 0-truncated objects, and the $(\infty,2)$-category of 1-localic $(\infty,1)$-toposes is equivalent to that of 1-toposes.

Jacob's answer points out that even among topological localizations of presheaves, many different $(\infty,1)$-toposes can have equivalent 1-toposes of 0-truncated objects, becuse we can consider $(\infty,1)$-toposes of sheaves on $(\infty,1)$-categories that are not 1-categories. (His examples, as you note, are categories of presheaves on $\infty$-groupoids.) This situation is entirely analogous to the 1-topos-theoretic one.

However, in the $\infty$-case there is the separate issue that even a left exact localization of a category of presheaves on a 1-category need not be 1-localic, if the localization is not topological. For a fixed 1-site, the lattice of corresponding localizations of its $(\infty,1)$-presheaf category, with the topological localization at one extreme and its hypercompletion at the other extreme, is probably what you are thinking of when you talk about "different ways of doing homotopy theory in the internal logic of $X$". So in this sense, $(\infty,1)$-toposes are more of a generalization of 1-toposes than 1-toposes are of locales.

(Incidentally, I'm not sure how accurate it is to think of the various localizations of presheaves on the site of a 1-topos $X$ as "ways of doing homotopy theory in the internal logic of $X$". As far as I know, only the hypercomplete such localization can actually be constructed "internally" to the 1-topos $X$, using the model structure on simplicial sheaves. It seems more accurate to me to describe the others as "ways of extending the internal logic of $X$ to include homotopy theory". But I'd be interested to hear others' thoughts on that subject.)