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Stream: learning: questions

Topic: Unknown property of posets

Ralph Sarkis (Jun 16 2022 at 15:38):

I have stumbled onto the following property of posets and I don't know its name or whether it has a name.

Definition:
A complete poset $P$ has property $*_\kappa$ if for every subset $S\subseteq P$, there exists a subset $S_\kappa \subseteq S$ of cardinality at most $\kappa$ satisfying $\sup S_\kappa = \sup S$.

Examples:

• Any complete poset $P$ has property $*_{|P|}$.
• The extended real numbers $\R \cup \{-\infty,\infty\}$, or any interval $[a,b]$, with the usual order has property $*_{\aleph_0}$.
• The natural numbers with the opposite order ($\sup$ is now $\inf$) has property $*_1$.
• I am guessing it would be a good exercise in combinatorics to figure out the smallest $\kappa$ such that the poweset of $S$ with inclusion order has property $*_\kappa$.

This property may be interesting or not (my use case is not particularly exciting), but I am asking here because of the second part of my question. I am in the process of learning more categorical logic/algebra so I tried to generalize this property to categories as an expert in algebraic categories would do and I wanted to confirm my guess.

A category $C$ has property $*_\kappa$ if for any filtered diagram $D:J \to C$, there is a filtered subdiagram $D_{\kappa}: J_{\kappa} \to C$ of cardinality at most $\kappa$ satisfying $\mathrm{colim}D_{\kappa} = \mathrm{colim}D$.

(Maybe I should assert that $\kappa$ is regular and say $D_\kappa$ has cardinality less than $\kappa$.)

Zhen Lin Low (Jun 16 2022 at 15:43):

For a given $\kappa$, let $J$ be the cardinal successor of $\kappa$ considered as a category. Then $J$ is filtered but has no cofinal subcategory of cardinality $\kappa$ or less . So, for example, $C = \textbf{Set}$ does _not_ have the property you define, for any $\kappa$.