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Stream: learning: questions

Topic: Axiomatizing Connected Components

John Onstead (Mar 17 2026 at 10:06):

In an extensive category like Top, there is a natural notion of connectivity in terms of objects whose homs preserve coproducts. But in Top, there's many other notions of connectivity- arc connectedness, path connectedness, simply connected, etc. In fact, there might be a notion of connectivity for every homotopy level, which would therefore imply an infinite number of notions of connectivity within just Top. As such, connectivity cannot merely be extra property on a category- it must be extra structure! The question is then: how might one axiomatize this, IE, "A 'category with connected objects' is a category C, equipped with this and that structure, such that these conditions hold..." Here's my attempt:

John Onstead (Mar 17 2026 at 10:07):

Definition: A category with connected objects is a pair $(C, D)$ where $C$ is a category with strict initial object, equipped with a subcategory $D$ of "connected objects", such that the following axioms hold:

Let $D(X)$ be the subobject poset of some object $X$ on all subobjects whose domain lies in $D$ . Then $D(X)$ is a forest poset- that is, a disjoint union of connected posets where each one has a unique maximal element or "root". Such a root subobject is called a "connected component".
Every morphism from an object in $D$ into any $X$ must factor through one, and only one, connected component of $X$ .
$D$ does not contain the strict initial object, so the functor $D/(-): C \to \mathrm{Cat}$ sending an object to the category of arrows from $C$ into it preserves the strict initial object.
$D/(-)$ reflects and preserves disjoint pullbacks.

So basically we have a system where every object has well-defined connected components, a connected object must always map (as a generalized element) into some single connected component (and so by composition connected components cannot be "split" by morphisms in the category), and where generalized elements being in different connected components implies that they are disjoint within $C$ 's inherent notion of disjointness.

John Onstead (Mar 17 2026 at 10:07):

My question is: true or false for each of the following three criteria, does the above axiomatic system for axiomatizing connected components and objects succeed? Let me know what you think!

Is it self-consistent and well-posed (the bare minimum requirements)?
Does it lack redundancy (that is, it's stated in simple terms and has no unnecessary components)?
Does it actually succeed to capture the intuition behind the idea of a "connected component" and recover common examples of things we would think of as being connected components? (Or is it either too weak or too strong in some regard to do so?)

David Egolf (he/him) (Mar 17 2026 at 17:09):

I'm wondering what some examples of categories with connected objects might be. Having some examples might be helpful for thinking about these questions!

If $C$ is an extensive category with a strict initial object, and $D$ is its full subcategory of connected objects (in the sense of the linked nLab article), then is $(C,D)$ a category with connected objects? For example, one might contemplate the case where $C$ is $\mathsf{Top}$ , the category of topological spaces.

John Onstead (Mar 17 2026 at 22:05):

David Egolf (he/him) said:

If $C$ is an extensive category with a strict initial object, and $D$ is its full subcategory of connected objects (in the sense of the linked nLab article), then is $(C,D)$ a category with connected objects?

That's exactly one of the things I want to figure out!

John Onstead (Mar 17 2026 at 22:06):

For the Top case, it is true. I had how connected objects work in Top in mind when I was writing the axioms.

John Onstead (Mar 17 2026 at 22:12):

Here's an interesting fact I was able to prove: Let $(C, D)$ be a category with connected objects. Then if $C$ furthermore has coproducts, and these coproducts are disjoint, then $\mathrm{Coprod}(D)$ (the coproduct closure of $D$ in $C$ ) is a coreflective subcategory of $C$ .

Basically, given an object $X$ , we can form the coproduct of its connected components $\coprod X_i$ and the canonical map by the universal property of the coproduct $\coprod X_i \to X$ . It's then possible to show that this is the universal morphism from $\mathrm{Coprod}(D)$ to $X$ , meaning any other map $Y \to X$ where $Y \in \mathrm{Coprod}(D)$ factors uniquely through it. I'm currently working to show if the converse is true.

David Egolf (he/him) (Mar 18 2026 at 03:19):

John Onstead said:

Basically, given an object $X$ , we can form the coproduct of its connected components $\coprod X_i$ ...

This is a cool idea! (And your post inspired me to learn more about coreflective subcategories, too. So that was fun!)

David Egolf (he/him) (Mar 18 2026 at 03:26):

I'm reminded that in a locally connected topos, we have for any object $X$ the following: there exist some $X_i$ so that $X \cong \coprod_i X_i$ , where each $X_i$ is a connected object.

It's a bit mind-bending (for me, at least) to imagine a situation where sticking together all the connected components of an object can give us something different from what we started with!

David Michael Roberts (Mar 18 2026 at 03:39):

Please note that for general topological spaces, you can't form [the given space as] the coproduct of the connected components! eg for totally disconnected spaces, the connected components are singletons, but they aren't discrete...

David Michael Roberts (Mar 18 2026 at 03:40):

So for a general space $X$ , $\pi_0(X)$ naturally has a totally disconnected topology, and $X\to \pi_0(X)$ is the universal map to a totally disconnected space.

David Michael Roberts (Mar 18 2026 at 03:42):

Where here $\pi_0$ denotes connected components.

You can tell the same story for path components, and the universal map to a totally path-disconnected space (i.e. one where every continuous function from an interval is constant).

John Baez (Mar 18 2026 at 04:28):

What do you mean, "you can't form the coproduct of the connected components"? Top has coproducts. (I agree that a topological space ain't in general the coproduct of its connected components.)

David Michael Roberts (Mar 18 2026 at 04:48):

Sorry, I meant "you can't form the given space as the coproduct of its connected components", yes.

John Baez (Mar 18 2026 at 04:55):

Got it. Just for @David Egolf (he/him), who said

It's a bit mind-bending (for me, at least) to imagine a situation where sticking together all the connected components of an object can give us something different from what we started with!

let me point out that the rational numbers with their usual topology as a subspace of $\mathbb{R}$ is a situation like this.

When you shatter a space like this into a dust of points, all the king's horses and all the king's men can't put Humpty together again - not as a coproduct of points, anyway.

Gabriel Goren Roig (Mar 18 2026 at 21:37):

Hey @John Baez, your last sentence made me think about the following. Clearly any colimit of points must be a discrete space, because the point has no (non-trivial) automorphisms. In other words, no colimit of points can recover a non-discrete space. But could there be some space which is not the coproduct of its connected components and yet can be recovered as some other colimit of those components? The answer is also no. A topological space is a colimit of connected spaces if and only if it is a coproduct of connected spaces, because the full subcategory of connected spaces $\mathscr{C} \hookrightarrow \mathsf{Top}$ is closed under connected colimits. Therefore in any diagram of connected spaces we can replace every connected component of the diagram with a single connected space. We conclude that an arbitrary topological space cannot be recovered as any colimit of its connected components (even if these components are not points).

This is related to the fact that the density comonad of the inclusion $\mathscr{C} \hookrightarrow \mathsf{Top}$ can be computed pointwise as $X\mapsto \coprod_{C \subseteq X\text{ conn. comp.}} C$ (i.e. it is precisely the operation on spaces that you were discussing) and is therefore an idempotent comonad. Equivalently, spaces that are coproducts of their connected components are coreflective in $\mathsf{Top}$ .

Without being a topologist, I wonder whether this explains a sense in which $\mathsf{Top}$ is said to be badly behaved: the behavior of things "touching each other without actually coinciding" (I guess the simplest such example would be the acumulation point of $Y = \{1/n \mid n \in \mathbb{N}\} \cup \{0\}$ with the subspace topology) cannot be captured or "generated" using colimits; it must be assumed as part of the building blocks. On the other hand, interestingly this can be captured with limits, not colimits ( $Y$ is a cofiltered limit of discrete finite spaces, as is any Stone space or so they say). So perhaps starting from a single point you can take colimits to get discrete spaces, then limits, then colimits again and you get a much broader class of spaces in this way. (Is this what condensed mathematics is about?)

John Onstead (Mar 18 2026 at 22:37):

Gabriel Goren Roig said:

Equivalently, spaces that are coproducts of their connected components are coreflective in $\mathsf{Top}$ .

This was one of the motivations for me to prove coreflectivity in my "category with connected components" construction.

John Onstead (Mar 18 2026 at 22:37):

Here's an interesting thing about this. For a while I was calling objects in $\mathrm{Coprod}(D)$ , the coproduct closure of $D$ a subcategory of connected objects, as "locally connected objects". But in Top, these "locally connected objects" do not converge with the topological notion of locally connected space! For instance, the topologist sine curve is trivially locally connected in the category theory sense (since it is a connected object), but it is not locally connected in the topological sense.

John Onstead (Mar 18 2026 at 22:42):

David Michael Roberts said:

So for a general space $X$ , $\pi_0(X)$ naturally has a totally disconnected topology, and $X\to \pi_0(X)$ is the universal map to a totally disconnected space.

That makes me wonder when a "category with connected components" as defined by me above admits a notion of "totally disconnected object", and when it furthermore admits the totally disconnected objects as a reflective subcategory of $C$ . More things for me to think about on this topic!

David Michael Roberts (Mar 19 2026 at 00:16):

Gabriel Goren Roig said:

Clearly any colimit of points must be a discrete space, because the point has no (non-trivial) automorphisms.

Note that if instead you use compact Hausdorff [[extremally disconnected space]] s instead of (finite sets of) discrete points, then you can get eg arbitrary compact Hausdorff spaces under taking colimits. So there's only a very small amount of 'glue' you need between your ingredients, to get very connected up spaces eg spheres, closed Riemann surfaces etc. This is part of the magic that goes into condensed mathematics.

John Baez (Mar 19 2026 at 02:19):

Can you say how to get the real line as a colimit of compact Hausdorff extremally disconnected spaces? Is this black magic using the axiom of choice (like ultrafilters), or is it something that kids can do at home?

David Michael Roberts (Mar 19 2026 at 03:35):

There is a quotient map from the Cantor set to the interval [0,1], for instance, sending a binary expansion (taken as an element of the set {0,1}^|N with the product topology) to the actual real number... Repeat by gluing all the intervals together to get the real line

David Michael Roberts (Mar 19 2026 at 03:37):

You can do the same thing with the decimal expansion, the infinite product of {0,...,9} is also a extremally disconnected. Scholze talks about it in this talk https://www.youtube.com/watch?v=uHJizsbVmhk

David Michael Roberts (Mar 19 2026 at 03:39):

It's kind of weird, but also gives an interesting insight into the struggle people have with the intuition that 0.999999... = 1.00000..... They are different points in the totally disconnected space of decimal expansions, but are identified under a quotient to the connected real line. You cannot continuously move in this non-connected space from any truncation of 0.999999.... to 1.0000..., but people are thinking of the real numbers as a continuum.

John Baez (Mar 19 2026 at 04:02):

David Michael Roberts said:

There is a quotient map from the Cantor set to the interval [0,1], for instance, sending a binary expansion (taken as an element of the set {0,1}^|N with the product topology) to the actual real number... Repeat by gluing all the intervals together to get the real line.

Oh, right! Great. Thanks.

Morgan Rogers (he/him) (Mar 23 2026 at 08:35):

David Michael Roberts said:

So for a general space $X$ , $\pi_0(X)$ naturally has a totally disconnected topology, and $X\to \pi_0(X)$ is the universal map to a totally disconnected space.

Correction: this should be a universal map $\pi_0(X) \to X$ .

Graham Manuell (Mar 23 2026 at 11:31):

Morgan Rogers (he/him) said:

Correction: this should be a universal map $\pi_0(X) \to X$ .

No, it was the right way before.

Morgan Rogers (he/him) (Mar 23 2026 at 11:36):

What is the map from $\mathbb{Q}$ to $\pi_0(\mathbb{Q})$ ?

Morgan Rogers (he/him) (Mar 23 2026 at 11:39):

I was reading $\pi_0(X)$ as "the coproduct of the connected components" in the context of David's message, but I suppose from the notation that it was actually meant to refer to the quotient of $X$ identifying points in sharing a connected component (in which case I take back my correction but think that clarification was needed!)

John Baez (Mar 23 2026 at 19:22):

Morgan Rogers (he/him) said:

What is the map from $\mathbb{Q}$ to $\pi_0(\mathbb{Q})$ ?

For any space there's a function sending each point to the connected component it's in, but this is not always continuous if we give $\pi_0$ the discrete topology. I think David is using the quotient topology.

David Michael Roberts (Mar 24 2026 at 05:03):

Morgan Rogers (he/him) said:

I was reading $\pi_0(X)$ as "the coproduct of the connected components" in the context of David's message, but I suppose from the notation that it was actually meant to refer to the quotient of $X$ identifying points in sharing a connected component (in which case I take back my correction but think that clarification was needed!)

$\pi_0(X)$ is not the coproduct of connected components. The map I was mentioning is the unit for the adjunction coming from the fact totally disconnected spaces are a reflective subcategory in Top. When I said " $\pi_0(X)$ naturally has a totally disconnected topology", I meant it, otherwise I would have said "discrete topology".

David Michael Roberts (Mar 24 2026 at 05:04):

I had written a comment before Graham had replied, but for some reason it didn't post. It was roughly about considering the case that $X=S^1 \sqcup S^1$ , so that $\pi_0(X)\simeq \{0,1\}$ with the discrete topology. What is the universal map $\{0,1\} \to X$ ?

Morgan Rogers (he/him) (Mar 24 2026 at 09:02):

If $\pi_0(X)$ had meant the coproduct of connected components you were talking about immediately before bringing it up, then it would be the identity map!
Look, I know it's silly to have proposed a correction considering I know very well what $\pi_0$ usually means, but apparently the passage from the coproduct of connected subobjects (that was actually under discussion) to $\pi_0$ short-circuited me.