# Sharply less regular cardinals in set theory

If $\lambda<\kappa$ are two regular cardinals, then every locally $\lambda$-presentable category is locally $\kappa$-presentable. The analogous statement does not hold for $\lambda$-accessible categories, though. In the books by Adamek-Rosicky and Makkai-Paré a special relation $\lambda\vartriangleleft \kappa$ for this property is introduced. It holds if and only if for every set $X$ with $\# X < \kappa$ the partial order $P_{<\lambda}(X)$ of subsets of $X$ with $<\lambda$ elements has a cofinal subset of cardinality $<\kappa$.

Several properties are established (see also MO/150305) which help to study accessible categories. But in principle this relation between regular cardinals is a purely set theoretic concept. Therefore, I wonder if set theorists have actually considered to use this relation, and which applications outside of category theory and categorical logic are there.

• There is a rich literature concerning generalized covering and dominating numbers, or generalized cardinal characteristics, to which this concept is related. For exampe, Levi's answer to the question to which you link points at one article. – Joel David Hamkins Dec 14 '16 at 13:59
• I think this is basically what I was asking here. – Tim Campion Dec 14 '16 at 16:00
• @Tim: Right! I didn't see see your question. Perhaps my question then can get closed as a duplicate. On the other hand, your question has no detailed answer. – HeinrichD Dec 14 '16 at 19:42