$\infty$-topoi versus condensed anima

Question

Let $ExDisc_\kappa$ denote the category of $\kappa$-small extremally disconnected topological spaces (for now fix a strong limit cardinal $\kappa$). There's a functor $ExDisc_\kappa \to \mathsf{RTop}$ (the $\infty$-category of $\infty$-topoi and geometric morphisms) given by $X\mapsto Sh(X)$ and the pushforward.

It preserves coproducts (because the opposite $ExDisc_k^{op} \to \mathsf{LTop} \to Cat_\infty$ preserves products, and $\mathsf{LTop} \to Cat_\infty$ preserves and thus reflects small limits).

Therefore, if I left Kan extend it to $\kappa$-condensed anima (which are the free sifted cocompletion of $ExDisc_k$ - let me use the word "anima" to be sure not to confuse with topological spaces), I get a functor $Cond_\kappa \to \mathsf{RTop}$ which preserves all colimits.

Alternatively, its opposite $Cond_\kappa^{op} \to \mathsf{LTop}$ preserves all limits, and those are computed in $Cat_\infty$. I think it follows that this opposite functor, when restricted to discrete (but not static - so by discrete I mean in the sheaf-theoretic sense, in other words "constant") condensed anima is just $X \mapsto Fun(X,An)$; and for a wide class of $\kappa$-compactly generated topological spaces $X$ (I'm not sure what the technical assumptions are, here - I'll write CG for "compactly generated"), it should be $Sh(X)$ . In particular, this functor is fully faithful on discrete anima, and on a large class of static $\kappa$-CG topological spaces (I might want to add the word "sober" here to specify further the class of topological spaces to which this applies - static is to emphasize that there is no "homotopical dimension" ).

Now let $\kappa$ vary and we get a functor $S:Cond\to \mathsf{RTop}$ which is fully faithful on discrete anima and on a wide class of CG-topological spaces.

Furthermore, I think Clausen and Scholze have proved that for a suitably nice CG-topological space (again, I forget the hypotheses) $X$, there is an initial discrete anima $hX$ with a map $X\to hX$ in $Cond$; and they proved that for $X$ nice enough (probably a CW-complex), this $hX$ is the homotopy type of $X$. Now Mike Shulman seems to state something similar about $\infty$-topoi here (paragraph starting with "The really lovely observation"). This seems to suggest that the map $X\to hX$ in $Cond$ is sent to the map Shulman describes in $\mathsf{RTop}$.

Q1 : is it the case, i.e. can one identify these two maps ?

If these maps are indeed the same, then this shows that $map(X,Y)\to map(S(X),S(Y))$ is an equivalence when $X$ is a discrete anima and $Y$ too, or when $X$ is a nice enough CG-topological space and $Y$ is either a discrete anima or a similarly nice enough CG-topological space.

Now this is a wide class of objects and so one is led to wonder:

Q2 : is $S$ fully faithful ? If so, can one describe its essential image "intrinsically" ?

Q3: if not, is $S$ at least fully faithful on the full subcategory containing discrete anima and nice enough CG-topological spaces ? Can one describe a full subcategory of $Cond$ larger than the ones I described on which $S$ is fully faithful ?

Q4 : If yes to Q3 but not Q2, what is a good example of failure of fully faithfulness ? If no to Q3, what is a good example of a pair (discrete anima, nice enough CG-topological space) where the above map fails to be an equivalence ? (can one be found explicitly ?)

Motivation : it seems like $Cond$ and $\mathsf{RTop}$ achieve a similar "goal", which is to have a place where $\infty$-groupoids and "honest" topological spaces can interact. Furthermore, both have suitably nice fully faithful functors from both of these categories, and as I explained above, one can even define $Cond\to \mathsf{RTop}$ making the appropriate diagrams commute.

The existence of the map $X\to hX$ is one further similarity that makes one wonder what the difference between the two really is - especially when Shulman seems to describe (heuristically, there is no precise claim made in that blog post and so any misinterpretation is on my part) $\mathsf{RTop}$ as a sort of "pushout" of these two worlds.

Addendum : Maybe I should specify that I'm by no means an expert in either condensed stuff or (higher-)topos theory, so maybe I'm completely off here, but I'm hoping something can be said anyways.

Answer 1

So I think Dustin's comment and the linked comment thread answer most of the questions, if not all. I think it would nonetheless be good to have an account here so let me try to write a coherent answer here - all of it comes from the comment thread here, you should look for Dustin's comment on April 5 and the subsequent comments. All mistakes are mine.

A1: As pointed out in the comment thread, there is a canonical comparison map under $S(X)$ of the form $Psh(shape(S(X))) \to Psh(hX)$, which shows that if it is an equivalence, then the two maps in question are identified. In particular, if these two maps are equivalences, then by $2$-out-of-$3$ the latter must be too. This is the case in the nice situations I mentioned in my question, but not necessarily in general (as far as I can tell).

A2: No, $S$ is not even conservative, as the example $*\to B\mathbb R$ shows.

Let me say a bit more about this example. Here $\mathbb R$ denotes the static condensed anima associated to the topological space $\mathbb R$, which is then a group object in $Cond$ and so one can take its classifying object. Note that it follows that $\mathbb R\not\simeq *$ in $Cond$

Now $B\mathbb R = \mathrm{colim}_{[n]\in \Delta^{op}} \mathbb R^n$ and so $S(B\mathbb R)\simeq \lim_{[n]\in\Delta} S(\mathbb R^n)$. As Dustin pointed out in the comments, because $\mathbb R^n$ is contractible, the pullback functor $S(*)\to S(\mathbb R^n)$ is fully faithful and so the limit (which can be computed in $Cat_\infty$, so any descent data contains only objects that are pulled back from $S(*)$ at each level) is equivalent to the limit over $\Delta$ of $S(*)$, i.e. to $S(*)$.

One can also understand this from a construction of $S(X)$ as a topos of "étale" morphisms to $X$, but let me stick to the above version.

However $B\mathbb R$ is not equivalent to $*$, as can be seen e.g. from the equivalence $\Omega B\mathbb R \simeq \mathbb R$.

A3: This one was not answered in the comment thread but Simon Henry commented below how to answer it : it suffices to prove fully faithfulness for $X$ a discrete anima and $Y$ a static topological space. But then $Fun^{L,ex}(Sh(Y), Fun(X,An)) \simeq map(X,Fun^{L,ex}(Sh(Y),An))$ but $Fun^{L,ex}(Sh(Y),An)$ is the set of points of $Sh(Y)$, e.g. if $Y$ is sober it's just $Y$ (as a set), but in any case it's a set (I think the underlying set of the soberification of $Y$), so that this is $map(\pi_0(X), Fun^{L,ex}(Sh(Y),An)) \simeq Fun^{L,ex}(Sh(Y),Fun(\pi_0(X),An))$.

The same holds in $Cond$, so this proves the result.

A4 : See A2 for a good example of failure of fully-faithfulness. The answer to Q3 is yes, so one can't expect an example of the kind I mentioned.