On the cardinal arithmetic of accessible categories

Question

If $\lambda, \mu$ are regular cardinals, say that $\lambda \trianglelefteq \mu$ if $\lambda \leq \mu$ and $$\forall X, \, |X| < \mu \implies \mathrm{cf} (P_\lambda(X)) < \mu$$ Here $P_\lambda(X)$ is the poset of subsets of $X$ of cardinality $<\lambda$, and $\mathrm{cf}$ takes the cofinality of a poset.

Then $\trianglelefteq$ is an $\mathrm{Ord}$-directed partial order on regular cardinals which is strictly weaker than the ordering by size (by $\mathrm{Ord}$-directed, I mean that any set of cardinals has a $\triangleleft$-upper bound). For instance, $\aleph_1 \not \trianglelefteq \aleph_{\omega+1}$ but $\aleph_0 \trianglelefteq \mu$ for all $\mu \geq \aleph_0$.

This relation comes up in the theory of accessible categories, because the following are equivalent when $\lambda \leq \mu$ are regular cardinals:

• Every $\lambda$-accessible category is $\mu$-accessible.
• The category $\mathrm{Pos}_\lambda$ of $\lambda$-directed posets and embeddings [which is always $\lambda$-accessible] is $\mu$-accessible.
• For every $\lambda$-directed poset $P$, the $\lambda$-directed subsets are cofinal in $P_\mu(P)$.
• $\lambda \trianglelefteq \mu$.

A good reference is Adámek and Rosický, though these facts mostly go back at least to Makkai and Paré. The only facts I know how to use to determine whether $\lambda \triangleleft \mu$ are the following:

1. $\lambda \triangleleft \lambda^+$
2. If $\left(\alpha <\lambda, \beta < \mu \implies \beta^\alpha < \mu\right)$, then $\lambda \triangleleft \mu$ for the somewhat boring reason that for $|X| < \mu$ we have $|P_\lambda(X)| < \mu$ so that certainly $\mathrm{cf} (P_\lambda(X)) < \mu$.
3. If $\lambda < \mu$ and $\mathrm{cf} \mu \leq \lambda$, then $\lambda^+ \not \triangleleft \mu^+$.

Questions I basically would just like to see what the set theorists (and category theorists) around here might have to say about the relation $\triangleleft$. Is this by chance a well-studied relation? Should I expect the question of whether $\lambda \triangleleft \mu$ holds to be mostly determined, or mostly independent, given reasonable descriptions of $\lambda$ and $\mu$? Should I expect (1), (2), and (3) to pretty much determine what can be determined, or should I expect there's more to say? I suppose things are not too complicated under GCH, for example. Does having good control over the $\beth$ function in general give good control over the functions $\mathrm{cf}(P_\lambda(\beta))$?

A really basic question: is the quantity $\mathrm{cf}(P_\lambda(X))$ (weakly) monotonic in $|X|$?

Answer 1

For your really basic question at the end: Yes, the cofinality of $P_\lambda(X)$ is weakly monotone in $X$.

Specifically, if $X\subseteq Y$, then $\newcommand\cof{\text{cof}}\cof(P_\lambda(X))\leq\cof(P_\lambda(Y))$. The reason is that if $A\subset P_\lambda(Y)$ is cofinal in $P_\lambda(Y)$, then the set $A\upharpoonright X=\{a\cap X\mid a\in A\}$ is cofinal in $P_\lambda(X)$.

(Unfortunately, I don't know much about your relation $\trianglelefteq$.)