Inclusion modulo nonstationary

Abstract

A classical theorem of Hechler asserts that the structure $\left({\omega }^{\omega },{\le }^{\ast }\right)$ is universal in the sense that for any $\sigma$-directed poset P with no maximal element, there is a ccc forcing extension in which $\left({\omega }^{\omega },{\le }^{\ast }\right)$ contains a cofinal order-isomorphic copy of P. In this paper, we prove a consistency result concerning the universality of the higher analogue $\left({\kappa }^{\kappa },{\le }^S\right)$: Theorem. Assume GCH. For every regular uncountable cardinal $\kappa$, there is a cofinality-preserving GCH-preserving forcing extension in which for every analytic quasi-order Q over ${\kappa }^{\kappa }$ and every stationary subset S of $\kappa$, there is a Lipschitz map reducing Q to $({\kappa }^{\kappa },{\le }^S)$.

Slow filtrations made explicit in the LCC derivation