Fix (a prime $p$ and) a chromatic height $h$. Recall that the Bousfield-Kuhn functor $\Phi_h: \mathcal M_h^f \to Sp_{T(h)}$ is monadic, where $\mathcal M_h^f \subseteq Top_\ast$ is a certain subcategory of (sufficiently-connected) "monochromatic" spaces, and $Sp_{T(h)}$ is the telescopic localization of the category of spectra (and the monad on $Sp_{T(h)}$ is the monad for Lie algebras).

Now, I'm pretty sure that $\mathcal M_h^f$ is not a stable $\infty$-category (that's kind of the point!). On the other hand, my favorite kind of unstable presentable $\infty$-category is an $\infty$-topos, which $\mathcal M_h^f$ is not for the simple reason that it is pointed. This leaves me wondering:

Question 1: What "kind of $\infty$-category" is $\mathcal M_h^f$? What nice properties does it have, beyond being presentable?

Question 2: Is there a variant of $\mathcal M_h^f$ which forms an $\infty$-category of "unpointed monochromatic spaces"? In particular, is there such a beast which is an $\infty$-topos?

Question 3: In another direction, is there a good "pointed analog" of the notion of an $\infty$-topos? Most naively -- is there a characterization of those $\infty$-categories of the form $\mathcal E_\ast$ (i.e. pointed objects of $\mathcal E$) where $\mathcal E$ is an $\infty$-topos? (And does $\mathcal M_n^f$ have these properties?)

The last question may deserve to be pulled out as a separate question...


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