Consider the object classifier of the $\infty$-topos of $\infty$-groupoids. For the role it plays in homotopy type theory as the type of types, let’s denote it as $Type = \coprod_{[F]} B Aut(F)$, the coproduct of the automorphism ∞-groups of all (small) homotopy types $[F]$.

Now attempting to imitate the duality between the category of sets and the category of complete atomic Boolean algebras, we might consider the map taking an $\infty$-groupoid $A$ to $[A, Type]$. Similarly to how the internal Boolean algebra structure on $\mathbf{2}$ induces a Boolean algebra structure on $[X, \mathbf{2}]$, for a set $X$, the internal $\infty$-topos structure on $Type$ induces an $\infty$-topos structure on $[A, Type] = \infty-Grpd/A$, the slice $\infty$-topos.

Question: Just as $[X, \mathbf{2}]$ is a special kind of Boolean algebra, being complete and atomic, how can I characterise those $\infty$-toposes of the form $[A, Type]$?

The answer will probably involve Mike Shulman's suggestions to me: having a set of tiny generators and being a Boolean presheaf $\infty$-topos.

Further questions: with such a characterisation in hand, should we expect a duality between such $\infty$-toposes and the $\infty$-topos of $\infty$-groupoids? What would be the equivalent of Stone Space-Boolean algebra duality?