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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMay 14th 2017
• (edited May 14th 2017)

I have added the example of the rational numbers (here) at totally disconnected topological space

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeMay 14th 2017

Since the nLab’s view is that connected spaces are inhabited, I changed the opening sentence slightly. I also added to the examples the fact that totally disconnected spaces are closed under limits in $Top$. Also linked to the extensive article connected space in Related Concepts.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMay 14th 2017

Thanks, I didn’t cross check. That opening sentence was inherited all the way from rev 1

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeMay 14th 2017

I created a Properties section, adding a statement that totally disconnected spaces form a reflective subcategory of $Top$. I’ve just finished giving the details of a proof.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMay 14th 2017

Sorry, Todd, i got interrupted before I could catch that hmeomorphism.

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeMay 14th 2017

No problem, Urs.

Some of this stuff reminds me vaguely of fracture squares that people were discussing a lot a while back. Let $\lambda: Top \to TotDisconn$ be the reflector, i.e., the left adjoint to the embedding $TotDisconn \hookrightarrow Top$, so $\lambda(X) = X/\sim$ is the quotient space upon dividing by the equivalence relation “same connected component”. So there’s a unit map $X \to \lambda(X)$. On the other hand, the embedding $LocConn \hookrightarrow Top$ of locally connected spaces is a coreflective subcategory, where the coreflector takes a space $X$ to the space $\rho(X)$ with the same underlying set as $X$ but retopologized by a finer topology, where connected components of open $U$ in $X$ are open in $\rho(X)$. So there’s a counit map $\rho(X) \to X$.

Then there’s a “dolittle square” (pushout + pullback) in $Top$

$\array{ \rho(X) & \to & \Delta \pi_0 (X) \\ \mathllap{counit} \downarrow & & \downarrow \mathrlap{id} \\ X & \underset{unit}{\to} & \lambda(X) }$

where the top horizontal map is itself identified with a unit map $\rho(X) \to \Delta \pi_0 \rho(X)$ for a string of adjoints $\pi_0 \dashv \Delta \dashv \Gamma \dashv \nabla: Set \to LocConn$, and the right hand map is identified with a counit map $\Delta \Gamma \lambda(X) \to \lambda(X)$ for the adjoint string $\Delta \dashv \Gamma \dashv \nabla: Set \to Top$. I don’t think it’s quite the same as a fracture square of the type you guys were discussing, e.g., in number theory where the integers are fractured into local completions, because the thing being “fractured” ($X$) into connected components sits at a different point of the square, but the situation seems vaguely cohesively redolent of the type of thing you were discussing.

All of this may be idle chatter, but I’m throwing it out there in case anyone has something to add. (BTW I’m not 100% sure that square is a pushout – that’s an offhand guess.)

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeMay 15th 2017
• (edited May 15th 2017)

Todd, how is it that

the right hand map is identified with a counit map $\Delta \Gamma \lambda(X) \to \lambda(X)$

and yet in the diagram the right hand map is written $id: \Delta \pi_0(X) \to \lambda(X)$? Similarly, the description of the top horizontal map doesn’t seem to match the diagram.

• CommentRowNumber8.
• CommentAuthorTodd_Trimble
• CommentTimeMay 15th 2017

Yeah, the point is that there are multiple descriptions of the same map, taking place at different levels or contexts. The underlying set $\Gamma \lambda(X)$ of the space $\lambda(X)$ is the set of connected components $\pi_0(X)$; the ’$id$’ stands for the identity function which is a continuous function from the discretification $\Delta \Gamma \lambda(X)$ of $\lambda(X)$ to $\lambda(X)$ itself, and this is a counit map for the adjunction $\Delta \dashv \Gamma$. And for the top horizontal map (where the pullback may be taken more or less as defining $\rho(X)$), the spaces $X$ and $\rho(X)$ have the same set of connected components, $\pi_0(X) = \pi_0 \rho(X)$ canonically, and the top map may thus be identified with a unit map for the adjunction $\Delta \dashv \pi_0$.