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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.

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    $\begingroup$ Ah — the composition monoidal structure on $Fun^{lex,acc}(Spaces,Spaces)$ in fact coincides with the cocartesian monoidal structure. So every such functor admits a unique monad structure. This is also true in the 1-categorical setting, so Johnstone must be giving a description of arbitrary pro-sets. I wonder if he realized that… $\endgroup$ Commented Jul 28 at 18:17
  • $\begingroup$ Is the first question then equivalent to "is every pro-space the shape of a topos"? I'm confused because this fails even for pro-sets and 1-topoi. $\endgroup$ Commented Jul 29 at 17:27
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    $\begingroup$ I see the problem: it's not true that composition coincides with products of pro-objects. This is true if the second term of the composition preserves filtered colimits, but not in general. So cartesian monads should not be just pro-spaces. $\endgroup$ Commented Jul 29 at 17:35

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