@CWcx I only think of formal schemes having one point

@DenisNardin no worries. Thanks as usual. :)

In an $\infty$-topos with enough points, do Postnikov tower converge in the weak sense that for object $X$, the natural map $X \rightarrow \varprojlim X_{\leq n}$ is an equivalence?

I know it's enough to check it pointwise and that the truncation functors are also computed pointwise, but I'm not sure what to do with the inverse limit.

What are the dualizable objects in the symmetric monoidal $\infty$-category of presentable $\infty$-categories?

I have the strong impression there are very few, but I don't have a precise answer. I'd also like to know

@SaalHardali If the symmetric monoidal structure is cartesian product then the unit is the trivial terminal category 1. If X is dualizable the zig-zag equations for dualizability imply that there is a factorization of the identity functor on X as X --> 1 --> X. So only 1 is dualizable.

thats not the usual symmetric monoidal structure chosen- you want the one where Spaces is the unit, I think. And then I dunno...

@CharlesRezk And here I am saying maximal subgroupoid like a chump...

@SaalHardali, Presheaf $\infty$-categories are dualizable, where the dual of P(C) is P(C^{op}). After that there are also presentable $\infty$-categories which are retracts of presheaf $\infty$-categories, i.e., they are accessible localizations D \subseteq P(C) such that the inclusion is both a left and a right functor.

For example, take a compact Hausdorff space X, and for each U in X, consider the sieve on U consisting of those V \subseteq U such that the closure of V is contained in U. This collection of sieves doesn't satisfy the axioms of a Grothendieck topology, but regardless of that we can still talk about sheaves and cosheaves of spaces with respect to these sieves. In this case the $\infty$-category of sheaves is a retract of P(O(X)) and the category of cosheaves is a retract of P(O(X)^{op}), and I

see also the answer to mathoverflow.net/questions/252698/…