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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.

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    $\begingroup$ I think you can extract answers to your questions from the comments at this n-cafe thread: golem.ph.utexas.edu/category/2020/03/… . Search for my first comment and Peter Scholze's reply. $\endgroup$ Mar 29, 2021 at 19:28
  • $\begingroup$ @DustinClausen : thanks for the pointer, I had seen the headline of this thread but not the comments ! It does seem like you and Peter answer all the questions ! :) (I think you had already told me that you didn't want to think of them as the same thing, but I guess the explicit example there is always good to have in mind). Except maybe the first part of Q3, but your comments seem to indicate that it's the wrong sort of question $\endgroup$ Mar 29, 2021 at 19:44
  • $\begingroup$ @DustinClausen : for the $B\mathbb R$-example, I think one needs a slightly more precise description of $S(B\mathbb R)$. In the comment thread you explain that it's a certain full subcategory of $Cond_{/B\mathbb R}$, and I'm guessing the objects in this full subcategory correspond to (certain) group maps $\mathbb R\to Aut(F)$ for some discrete anima $F$ ? (and $Aut$ the internal $Aut$) $\endgroup$ Mar 29, 2021 at 20:02
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    $\begingroup$ @DustinClausen : right, but here $\mathbb R^n$ is viewed as a topological space, right ? There are sheaves of sets on $\mathbb R$ that aren't trivial $\endgroup$ Mar 29, 2021 at 20:18
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    $\begingroup$ Sorry, yes, I missed a part in the explanation and therefore said something wrong (that it's a constant diagram with value $S(\ast)$). Thanks for catching that. What I should have said was that if you look at descent data then by definition it's all pulled back from $S(\ast)$. Therefore the limit is the same if you levelwise pass to the full subcategories given by things pulled back from $S(\ast)$. Now that one is constant with value $S(\ast)$ by constractibility. $\endgroup$ Mar 29, 2021 at 20:28

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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.

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  • $\begingroup$ For $Q3$, the answer to $A1$ gives you a kind of answer. If you take the category of CW-complexes (seen as static topolgical space) and discrete anima then S is fully faithful on this: in each category separately you already know this, for map from space to anima this is $A1$ and map in the other direction factor through the $\pi_0$ of anima in both cases. I think you can get a more uniform class of objects by looking at things that are "étale $\infty$-groupoid in the category of CW-complexes". $\endgroup$ Mar 30, 2021 at 13:44
  • $\begingroup$ @SimonHenry : ah I was indeed wondering about anima to space, it is clear to me that it factors through $\pi_0$ in $Cond$, but not as clear in topoi (as I said, I don't know much about them). Would you have a reference (or a quick proof) ? $\endgroup$ Mar 30, 2021 at 13:48
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    $\begingroup$ Oh right, a left exact colimit preserving functor $Sh(X)\to Fun(Y,An)$ should be equivalent to a map $Y\to Fun^{L,ex}(Sh(X), An)$, but the latter is just a set (namely $X$). That's a good point, thank you ! $\endgroup$ Mar 30, 2021 at 13:49
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    $\begingroup$ Regardless it could still be an interesting question if you restrict to $\kappa$-condensed on one side and a suitable bound on $\infty$-topoi on the other side ! $\endgroup$ Mar 30, 2021 at 14:12
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    $\begingroup$ Yes there are set theoretic issue in that adjunction: in general given a space $X$ and an $\infty$-topos you have a proper class of morphism $X \to T$. So indeed technically speaking this adjunction does not exists. $\endgroup$ Mar 30, 2021 at 14:15

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