Higher Dimensional Cardinal Characteristics for Sets of Functions

TL;DR

None

Contribution

None

Abstract

Much recent work in cardinal characteristics has focused on generalizing results about $\omega$ to uncountable cardinals by studying analogues of classical cardinal characteristics on the generalized Baire and Cantor spaces $\kappa^\kappa$ and $2^\kappa$. In this note I look at generalizations to other function spaces, focusing particularly on the space of functions $f:\omega^\omega \to \omega^\omega$. By considering classical cardinal invariants on Baire space in this setting I derive a number of "higher dimensional" analogues of such cardinals, ultimately introducing 18 new cardinal invariants, alongside a framework that allows for numerous others. These 18 form two separate diagrams consisting of 6 and 12 cardinals respectively, each resembling versions of the Cicho\'n diagram. These ZFC-inequalities are the first main result of the paper. I then consider other relations between these cardinals, as well as the cardinal $\mathfrak{c}^+$ and show that these results rely on additional assumptions about cardinal characteristics on $\omega$. Finally, using variations of Cohen, Hechler, and localization forcing I prove a number of consistency results for possible values of the new cardinals.