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Stream: learning: questions

Topic: What exactly are countable limits?

Nathaniel Virgo (Jan 31 2025 at 14:57):

I'd like to understand exactly what's meant by a countable limit, or more generally a $\kappa$-ary limit for some cardinal $\kappa$. (For example, Chen's A universal characterisation of the category of standard Borel spaces says, among other things, that the category of standard Borel spaces has all countable limits, and I'd like to make sure I really understand what that means.)

As an initial guess, I'd imagine a countable limit is one where the indexing category has only a countable set of morphisms. But this is suspicious because having a countable set of morphisms isn't invariant under equivalence, and so I'm wondering if the concept might actually be more subtle than that.

Morgan Rogers (he/him) (Jan 31 2025 at 15:02):

Right, the diagram need only be "essentially countable" or even have a countable initial subcategory. These are equivalence-invariant versions of what it means to be countable, although note that there is no way to restrict to 'countably infinite', since it's not hard to produce a countably infinite category which is equivalent to a given finite one.

Nathaniel Virgo (Jan 31 2025 at 15:05):

I guess "essentially countable" means equivalent to one with a countable set of morphisms?

What's an initial subcategory? (If it means the obvious thing it seems like it would always be empty.)

Morgan Rogers (he/him) (Jan 31 2025 at 15:57):

It's an unfortunate clash of terminology: it means dual of [[final functor]]. Composition with such an initial functor $C \to D$ doesn't change the values of limits of functors $D \to E$.

Mike Shulman (Jan 31 2025 at 16:29):

It's not unrelated though: a one-object subcategory is an initial subcategory precisely if the one object is an initial object.

Nathaniel Virgo (Jan 31 2025 at 17:02):

I'm having trouble seeing why. If I unpacked the definition correctly, an initial subcategory $I\subseteq C$ is one where

• for each $c\in C$ there exists some morphism $i\to c$ from some object $i\in I$

• given any two such morphisms $f:i\to c$ and $f':i\to c$, there exists a morphism $i\to i'$ in $I$ making the evident triangle commute (this is wrong, fixing it - I'll probably work this out for myself once I do)

If $I$ only has one object this implies that there exists a morphism from that object to any other object, but I'm having trouble seeing why that morphism has to be unique in $C$. (Hints appreciated.)

Nathaniel Virgo (Jan 31 2025 at 17:08):

Hmm, fixing the thing that's wrong involves introducing a "zigzag of morphisms", which is kind of a funny thing to think about - I guess it will take me a bit of work to see why that fact about initial objects is true. (Hints still appreciated if there's an easy way to see it.)

Morgan Rogers (he/him) (Jan 31 2025 at 17:12):

iirc, the zigzag of morphisms lives in the domain, so if there is only an identity morphism there, only trivial zigzags can appear.

Nathaniel Virgo (Jan 31 2025 at 17:13):

Ah, if one-object subcategory means one object with only an identity morphism then I see it. I was trying to prove that the single object couldn't have any endomorphisms

Mike Shulman (Jan 31 2025 at 22:53):

Sorry, I should have been more clear. Yes, I meant a subcategory that is as a category equivalent to the terminal category.