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 ?
Best Answer
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).