# Is there an $\infty$-topos of monochromatic spaces?

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...