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Here is a conditional answer to the question. Consider the following Hypothesis: If $\mathcal Y$ is a sheaf $\infty$-topos, then for any set of objects $Y_0 \subset \mathcal Y$, there exists a small, …

Homotopy type theory (HoTT) gives a natural internal language for studying $\infty$-groupoids. Riehl and Shulman give an extension of HoTT which gives an analogous internal language for studying $\inf …

Here is another sort of constraint. I'll write $Sp(\mathcal C)$ instead of $Stab(\mathcal C)$. Claim: If $\mathcal A \simeq Sp(\mathcal X)$ for a nontrivial [1] $\infty$-topos $\mathcal X$, then for a …

The answer is no. That is, a conservative, exact left adjoint need not be comonadic, $\infty$-categorically. For a counterexample, recall Thm 6.7 of Bousfield's Localization of spectra with respect to …

Not an answer -- the question is very much open! But I think it's worth compiling together some of the observations made in the comments (this answer is community wiki; feel free to add, correct, chan …

In response to the edit which nails things down to a formal system involving cardinals $\kappa_{-1} < \kappa_0 < \kappa_{1/2} < \kappa_1$: I'm going to go out on a perhaps more ill-advised limb and pr …

Let me record this as an answer: Blomquist and Harper show that $\Sigma^\infty$ is comonadic on simply-connected spaces. It occurs to me that if your ultimate goal is to understand unstable homotopy …

I've played a little bit with this question in the past, and I don't have anything firm, but here are some thoughts: It seems to me that the characteristic properties of fields have a lot to do with …

To summarize the discussion in the comments: $Fun(\mathsf{FinTop},\mathsf{Top})$ classifies objects (Proof: since a left adjoint functor $Fun(\mathsf{FinTop},\mathsf{Top}) \to \mathcal E$ correspon …

Most directly, this fact generalizes to the whole Goodwillie tower, which are certainly "constructed in a similar way". I don't think it generalizes to all algebraic theories. However, I think one c …