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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 24th 2019
   • (edited Jul 24th 2019)
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   Author: DavidRoberts
   Format: MarkdownItexThe connected components functor $\pi_0\colon Top \to Set$ lifts through $TotDisconnTop \to Set$, giving $\pi_0(X)$ the quotient topology coming from the natural map $X\to \pi_0(X)$. One can also define a pro-set version of $\pi_0(X)$ by taking the diagram $\pi_0^{pro}(X)$ with objects the surjective continuous maps $q\colon X\to S_q$ where $S_q$ is a discrete space, and maps the obvious commutative triangles. The limit of this diagram in $Top$ is a pro-discrete space, a subspace of the topological product $\prod_q S_q$. As such it is a Hausdorff totally disconnected space. I imagine some loss of information has happened in passing from the pro-set to the limit (since the limit functor from pro-sets to spaces is apparently not full; at least the version for locales is not). Here are some obvious questions: * by the universal property for $\pi_0(X)$, there is a map $\pi_0(X) \to \lim\pi_0^{pro}(X)$. Is this map the Hausdorffification? * Is $\pi_0(X)$ always Hausdorff? (There are totally disconnected spaces that aren't) * Does $\pi_0(X)$ contain the 'same information' as $\pi_0^{pro}(X)$? * Can we recover the space $\pi_0(X)$ from $\pi_0^{pro}(X)$? (probably not)

   The connected components functor $\pi_0\colon Top \to Set$ lifts through $TotDisconnTop \to Set$, giving $\pi_0(X)$ the quotient topology coming from the natural map $X\to \pi_0(X)$. One can also define a pro-set version of $\pi_0(X)$ by taking the diagram $\pi_0^{pro}(X)$ with objects the surjective continuous maps $q\colon X\to S_q$ where $S_q$ is a discrete space, and maps the obvious commutative triangles. The limit of this diagram in $Top$ is a pro-discrete space, a subspace of the topological product $\prod_q S_q$. As such it is a Hausdorff totally disconnected space. I imagine some loss of information has happened in passing from the pro-set to the limit (since the limit functor from pro-sets to spaces is apparently not full; at least the version for locales is not). Here are some obvious questions:

   • by the universal property for $\pi_0(X)$, there is a map $\pi_0(X) \to \lim\pi_0^{pro}(X)$. Is this map the Hausdorffification?

   • Is $\pi_0(X)$ always Hausdorff? (There are totally disconnected spaces that aren’t)

   • Does $\pi_0(X)$ contain the ’same information’ as $\pi_0^{pro}(X)$?

   • Can we recover the space $\pi_0(X)$ from $\pi_0^{pro}(X)$? (probably not)

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 24th 2019
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   Author: DavidRoberts
   Format: MarkdownItexOne could think of this in the same light as Clausen and Scholze's "[Condensed mathematics](https://www.math.uni-bonn.de/people/scholze/Condensed.pdf)", which is done working in sheaves over the pro-étale site of the point, or equivalently small sheaves on the category of extremally disconnected spaces with the extensive pretopology.

   One could think of this in the same light as Clausen and Scholze’s “Condensed mathematics”, which is done working in sheaves over the pro-étale site of the point, or equivalently small sheaves on the category of extremally disconnected spaces with the extensive pretopology.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 24th 2019
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   Author: Todd_Trimble
   Format: MarkdownItexRe "is $\pi_0(X)$ always Hausdorff?" -- if you do take a totally disconnected non-Hausdorff space $X$, isn't then the quotient map $X \to \pi_0(X)$ a homeomorphism? In general, a quotient map $p: X \to Y$ is a homeomorphism if the underlying function is a bijection, no?

   Re “is $\pi_0(X)$ always Hausdorff?” – if you do take a totally disconnected non-Hausdorff space $X$, isn’t then the quotient map $X \to \pi_0(X)$ a homeomorphism? In general, a quotient map $p: X \to Y$ is a homeomorphism if the underlying function is a bijection, no?

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 24th 2019
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   Author: DavidRoberts
   Format: MarkdownItexBleh, of course... :-/

   Bleh, of course… :-/

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 25th 2019
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   Author: DavidRoberts
   Format: MarkdownItexSo here is some progress: $\lim \pi_0^{pro}(X)$ is zero-dimensional Hausdorff, and there are continuous maps $\pi_0(X) \to \pi_0^{quasi}(X) \to \lim\pi_0^{pro}(X)$, where the middle space is that of _quasi-components_, and is totally separated (such spaces are equal to their own $\pi_0^{quasi}$). I think the second map is injective, but not an embedding in general. The first map is surjective, but not injective in general. There are also Hausdorff totally separated spaces that aren't zero-dimensional.

   So here is some progress: $\lim \pi_0^{pro}(X)$ is zero-dimensional Hausdorff, and there are continuous maps $\pi_0(X) \to \pi_0^{quasi}(X) \to \lim\pi_0^{pro}(X)$, where the middle space is that of quasi-components, and is totally separated (such spaces are equal to their own $\pi_0^{quasi}$). I think the second map is injective, but not an embedding in general. The first map is surjective, but not injective in general. There are also Hausdorff totally separated spaces that aren’t zero-dimensional.

6. • CommentRowNumber6.
   • CommentAuthorTim_Porter
   • CommentTimeJul 25th 2019
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   Author: Tim_Porter
   Format: MarkdownItexThis reminds me somewhat of what Eduardo Dubuc was doing in the late 1990s. He was if I remember correctly using locales however.Ieke Moerdijk also worked on this sort of thing. Something along was published as `Localic Galois theory' by Dubuc in Advances, in 2003 although that was going off in another direction. May I ask why you want to bother with taking the limit as pro-sets work better and taking the limit does destroy information? Of course the answer may just be `curiosity'!

   This reminds me somewhat of what Eduardo Dubuc was doing in the late 1990s. He was if I remember correctly using locales however.Ieke Moerdijk also worked on this sort of thing. Something along was published as ‘Localic Galois theory’ by Dubuc in Advances, in 2003 although that was going off in another direction.

   May I ask why you want to bother with taking the limit as pro-sets work better and taking the limit does destroy information? Of course the answer may just be ‘curiosity’!

7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 25th 2019
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   Author: DavidRoberts
   Format: MarkdownItexComplete curiosity. I guess the main question I was wondering is whether the pro-set version has the same information as the topological space of connected components. Certainly the limit of the pro-set has less information, but it's not even clear how one would sensibly phrase a statement of this sort.

   Complete curiosity. I guess the main question I was wondering is whether the pro-set version has the same information as the topological space of connected components. Certainly the limit of the pro-set has less information, but it’s not even clear how one would sensibly phrase a statement of this sort.

8. • CommentRowNumber8.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 26th 2019
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   Author: DavidRoberts
   Format: MarkdownItexOne could also consider the limit of the pro-set in the category of locales. This would potentially give another invariant. I saw this on the lab somewhere, but I can't find it again! It said on the page that the limit functor wasn't full, so likely there is loss of information there as well.

   One could also consider the limit of the pro-set in the category of locales. This would potentially give another invariant. I saw this on the lab somewhere, but I can’t find it again! It said on the page that the limit functor wasn’t full, so likely there is loss of information there as well.

9. • CommentRowNumber9.
   • CommentAuthorTim_Porter
   • CommentTimeJul 26th 2019
   • (edited Jul 26th 2019)
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   Author: Tim_Porter
   Format: MarkdownItexi recall something along those lines, possibly in work by Marta Bunge, or Eduardo Dubuc, or Ieke Moerdijk or combinations of them. I have a hard copy in a box file under a bed upstairs here, but am not volunteering to find it now! A perhaps related paper is _The fundamental localic groupoid of a topos_ in JPAA by John Kennison. Marta Bunge wrote: Classifying Toposes and Fundamental Localic Groupoids but I note these look more at the $\pi_1$ analogue of your question.

   i recall something along those lines, possibly in work by Marta Bunge, or Eduardo Dubuc, or Ieke Moerdijk or combinations of them. I have a hard copy in a box file under a bed upstairs here, but am not volunteering to find it now! A perhaps related paper is The fundamental localic groupoid of a topos in JPAA by John Kennison. Marta Bunge wrote: Classifying Toposes and Fundamental Localic Groupoids but I note these look more at the $\pi_1$ analogue of your question.

10. • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 26th 2019
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    Author: DavidRoberts
    Format: MarkdownItexThanks. The connected components locale of the $\Pi_1$ would give the $\pi_0$ probably, but it's more the comparison between these things which is intriguing me at the moment.

    Thanks. The connected components locale of the $\Pi_1$ would give the $\pi_0$ probably, but it’s more the comparison between these things which is intriguing me at the moment.