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Stream: learning: questions

Topic: A preorder is to an antichain as a category is to what?

Keith Elliott Peterson (Feb 17 2021 at 08:39):

Is there a name for the set $A(x)=\{y\in\mathrm{Obj}(\mathcal{C})\mid \hom[x,y]\cong\hom[y,x]\cong\varnothing\}$ (what one might call the space-like objects in a categoy)? If not, would anti-representable be fine?

Likewise, could we not consider, for each object $x$ of $\mathcal{C}$, a subcategory $\mathcal{C}\parallel x$, with $A(x)$ the objects, and the morphisms those of $\mathcal{C}$ with source and target in $A(x)$, a sort of analog to over and under categories? What should such a subcategory be called? I want to suggest outer category of $x$.

I might now be getting ahead of myself here, but could we not use such a construction, along with the over and under category at $x$, to define what a Huet zipper of a category at $x$ is?

Morgan Rogers (he/him) (Feb 17 2021 at 09:45):

The title is misleading here, since that definition doesn't give an antichain when you apply it to a preorder.
Why do you say the objects in $A(x)$ are "space-like"? Certainly anti-representable would be a bad name: representables are presheaves (or copresheaves) on a category, not objects of the category itself.
All of that said, $A(x)$ is well-defined, and the resulting full subcategory $\mathcal{C}||x$ that you describe is essentially preserved under equivalences of categories. It doesn't look anything like an under or over category, but outer category doesn't seem like a bad name if you're using this concept. It's descriptive, at least.
As for the zipper, you'll have to be a lot more precise about what you have in mind.

Amar Hadzihasanovic (Feb 17 2021 at 10:34):

I can answer about the “spacelike”: he's thinking of the terminology in relativity and causal structures where two points are in a “spacelike” relation to each other if there's no causal connection between them (no “timelike curve” that connects them).

Keith Elliott Peterson (Feb 17 2021 at 20:11):

Amar Hadzihasanovic said:

I can answer about the “spacelike”: he's thinking of the terminology in relativity and causal structures where two points are in a “spacelike” relation to each other if there's no causal connection between them (no “timelike curve” that connects them).

Yes, this is what I had in mind.

Keith Elliott Peterson (Feb 17 2021 at 20:35):

Morgan Rogers (he/him) said:

The title is misleading here, since that definition doesn't give an antichain when you apply it to a preorder.

Are not preorders the $\mathrm{Bool}$-enriched categories, which can also be identified with the $\mathcal{P}(\{\bullet\})$-enriched categories?

Morgan Rogers (he/him) said:

Certainly anti-representable would be a bad name: representables are presheaves (or copresheaves) on a category, not objects of the category itself.

Fair enough. What would be a good descriptive name then?

Morgan Rogers (he/him) said:

As for the zipper, you'll have to be a lot more precise about what you have in mind.

I'm probably not the best person to explain this, but a (Huet) zipper is a data structure that is a modification of an existing data structure with immediate access to a specific object in the original data structure.

https://en.wikipedia.org/wiki/Zipper_(data_structure)#Alternatives_and_extensions

Martti Karvonen (Feb 17 2021 at 21:25):

Keith Peterson said:

Morgan Rogers (he/him) said:

The title is misleading here, since that definition doesn't give an antichain when you apply it to a preorder.

Are not preorders the $\mathrm{Bool}$-enriched categories, which can also be identified with the $\mathcal{P}(\{\bullet\})$-enriched categories?

Yes, but for a preorder your set $A(x)$ is the set of elements incomparable with $x$: members of it might not be pairwise incomparable so the end result is not always an antichain.

John Baez (Feb 17 2021 at 22:09):

So, you might call $A(x)$ the set of objects incomparable to $x$.

John Baez (Feb 17 2021 at 22:10):

Or maybe disconnected from $x$.

John Baez (Feb 17 2021 at 22:11):

But maybe this concept doesn't have a name because nobody has figured out a really good use for it.

John Baez (Feb 17 2021 at 22:12):

So you might try to figure out something to do with it.