Let $\mathcal E$ be an $\infty$-topos, and let $t : \mathcal E \to Spaces$ be the unique geometric morphism to $Spaces$. The composite $t_\ast t^\ast : Spaces \to Spaces$ is a left-exact accessible functor, i.e. a pro-space, known as the shape of the $\infty$-topos $\mathcal E$.
Moreover, though, $t_\ast t^\ast$ is obviously a monad, a fact which I haven’t seen discussed in the $\infty$-categorical literature. The analogous monads have been studied in the 1-categorical literature: in Cartesian monads on toposes, Johnstone shows that every monad on $Set$ whose underlying endofunctor is left exact arises as the shape of a topos. He moreover shows that the functor $\mathcal E \mapsto t_\ast t^\ast$, $Topos \to CartMnd(Set)$ is a localization, identifying a topos $\mathcal E$ with its “strongly zero-dimensional localic” reflection [1]. This leads to an obvious
Question 1: How much of this story lifts to $\infty$-topoi?
Does every monad $T$ on $Spaces$ whose underlying endofunctor is left-exact arise as $T = t_\ast t^\ast$ for some $\infty$-topos $\mathcal E$?
Is the resulting “enhanced shape” $\infty-Topos \to CartMnd(Spaces)$ a localization?
Can we characterize these possible “enhanced shapes” in a way analogous to the “strongly zero-dimensional locales” of Johnstone?
Question 2: Shapes of $\infty$-topoi are used to do real work (see e.g. etale homotopy theory, exodromy, ... ). Can we get more mileage out of $\infty$-topos-theoretic shape theory by considering this monad aspect?
[1] A topological space is “strongly zero-dimensional” if it is zero-dimensional and moreover every open cover has a pairwise-disjoint open refinement. Every compact (even Lindelof) zero-dimensional space is strongly zero-dimensional, but there are noncompact ones and not every zero-dimensional space is strongly zero-dimensional.
