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](https://ncatlab.org/nlab/show/shape+of+an+%28infinity%2C1%29-topos) 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](https://doi.org/10.1016/S0022-4049(96)00165-X), 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](https://ncatlab.org/nlab/show/%C3%A9tale%2520homotopy), [exodromy](https://arxiv.org/abs/1807.03281), ... ). 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.