Higher walks and squares

TL;DR

This paper advances the theory of higher dimensional walks on ordinals, introducing new coherence and square principles, analyzing their properties in the constructible universe, and linking them to topological cohomology.

Contribution

It develops the theory of higher dimensional walks, defines new higher square principles, and explores their consistency and topological implications.

Findings

Higher square principles exist in the constructible universe.

These principles can be forced to fail using large cardinals.

Certain higher rho-functions show non-triviality and coherence.

Abstract

We continue the development of the theory of higher dimensional walks on ordinals began recently by Bergfalk. In particular we identify natural coherence conditions on higher dimensional $C$-sequences that entail coherence of the resultant higher rho-functions. We also introduce various higher square principles by adding non-triviality conditions to these coherent higher $C$-sequences and investigate basic properties of said square principles. For example, in analogy with the classical case, we prove that these higher square principles abound in the constructible universe but can be forced to fail, modulo large cardinals. Finally, we prove that certain higher rho-functions obtained by walking along higher square sequences exhibit non-triviality in addition to coherence. In particular, it follows that higher square principles on a cardinal $\lambda$ entail certain non-vanishing \v{C}ech cohomology groups for $\lambda$ considered with the order topology.