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Stream: learning: questions

Topic: objects as colimits of points

Jules Hedges (Nov 08 2021 at 15:02):

In many categories, every object is a colimit of copies of the terminal object (unless this entire question is mistaken, which is for sure possible). Expanding that a bit, fixing a suitable category $\mathcal C$: for every object $X$ there exists a small category $J_X$ such that $X$ is the colimit of the terminal functor $J_X \to \mathcal C$.
1) Is there some theory of this process? Under what conditions on $\mathcal C$ can you always do this?
2) Is there ever any sense in which the association $X \mapsto J_X$ is "canonical"? Or are there in general just a whole bunch of unrelated $J_X$ and no way to pick one?
3) How do you go about actually computing these things in practice? Let's say, to pick a random example, that I want to get $\mathbb R$ as a colimit of points in $\mathbf{Top}$. How would I go about finding a category $J_\mathbb R$ that does what I want?

Reid Barton (Nov 08 2021 at 15:10):

Only discrete spaces are colimits of points in Top. I think there are not many categories with this property (if you want them to have all colimits)--maybe just Set and its two localizations. What does happen often is that there is some good way to write objects of C as colimits of a fixed small collection of objects of C.

Jules Hedges (Nov 08 2021 at 15:14):

Jules Hedges said:

(unless this entire question is mistaken, which is for sure possible)

Huh.

Jules Hedges (Nov 08 2021 at 15:14):

Until roughly 2 minutes ago I would have said that every space is a quotient of a discrete space without thinking about it

Zhen Lin Low (Nov 08 2021 at 15:16):

It is true that every ∞-groupoid is a quotient of discrete set... but that's a different notion of space!

Joe Moeller (Nov 08 2021 at 15:18):

Here's a minimal example. The Sierpinski space (the set {1,2} with the topology {{}, {2}, {1,2}}) I believe does not admit a quotient map from any discrete space.

Zhen Lin Low (Nov 08 2021 at 15:20):

The quotient of any discrete topological space is again a discrete topological space. I would say the 2-point indiscrete space is the minimal counterexample, but I suppose the Sierpiński space is minimal if you restrict to sober spaces.

Reid Barton (Nov 08 2021 at 15:22):

Exactly, it's not totally obvious because there are more possible indexing categories than just discrete ones, but you can check that a colimit of discrete spaces is discrete by using the fact that a space is discrete iff any map from it to the discrete space on 2 elements is continuous.

Reid Barton (Nov 08 2021 at 15:22):

For the original question I guess there are also Kleisli categories of monads on Set that preserve 1, e.g., free convex spaces or spaces of the form $\beta S$ ($\beta$ = the ultrafilter monad)

Jules Hedges (Nov 08 2021 at 15:26):

I've been working recently with categories where every object is a coproduct of points. I was (also mistakenly) thinking that any kleisli category of a monad on $\mathbf{Set}$ is an example of that, although now what I think is actually true is that every object is a coproduct of copies of the free algebra on 1 generator, which is terminal I think exactly when the monad is affine (ie. $T 1 = 1$). An interesting example is the kleisli category of the finite support probability distribution monad, aka the category of finite support Markov kernels

Jules Hedges (Nov 08 2021 at 15:27):

That's why I've been wondering how to make that kind of trick work in general

Reid Barton (Nov 08 2021 at 15:36):

Oh yeah these aren't counterexamples to my original guess because these categories lack other colimits--then you could also include examples such as finite sets, or infinite sets, etc.

Reid Barton (Nov 08 2021 at 15:36):

Well not infinite sets.

Mike Shulman (Nov 08 2021 at 16:42):

Jules Hedges said:

2) Is there ever any sense in which the association $X \mapsto J_X$ is "canonical"? Or are there in general just a whole bunch of unrelated $J_X$ and no way to pick one?

If the inclusion of $\{1\}$ is a [[dense functor]] then there is by definition a canonical choice.

John Baez (Nov 08 2021 at 18:57):

Jules Hedges said:

In many categories, every object is a colimit of copies of the terminal object (unless this entire question is mistaken, which is for sure possible).

What are some categories like that? I'm having trouble thinking of categories like that, other than Set (and FinSet, and similar categories).

My mental image of a terminal object is a point, and when I take a bunch of points and glue them together however I want, I just get a bunch of points. That is: in my mind's eye, any colimit of terminal objects is just a coproduct of terminal objects, which is just some number of dots - possibly infinite.

John Baez (Nov 08 2021 at 19:01):

There could be categories where taking colimits of terminal objects gives more interesting objects, but I'd like to hear about them!

Mike Shulman (Nov 08 2021 at 19:05):

I'm sure this is not what Jules had in mind, but of course in a higher category one can get interesting things by gluing together points because the glue retains its shape. For instance, $S^1$ is the (homotopy) colimit of the diagram $1\rightrightarrows 1$.

Jules Hedges (Nov 09 2021 at 10:53):

John Baez said:

What are some categories like that? I'm having trouble thinking of categories like that, other than Set (and FinSet, and similar categories).

My mental image of a terminal object is a point, and when I take a bunch of points and glue them together however I want, I just get a bunch of points. That is: in my mind's eye, any colimit of terminal objects is just a coproduct of terminal objects, which is just some number of dots - possibly infinite.

I was mistaken, I had in mind things like Top but it's not true. (There are at least some other examples besides Set and similar things, you need to cheat a bit by looking at categories that only have some colimits: for example, the kleisli category of any affine monad on Set has every object being a coproduct of points. But the spirit of what I had in mind was to go beyond examples like that)