Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Higher walks and squares

Chris Lambie-Hanson, Pedro Marun

TL;DR

The paper advances higher-dimensional analogues of Todorcevic's walks on ordinals by introducing coherent $n$-$C$-sequences and higher square principles, and by defining enriched higher-walk constructs that yield coherent yet nontrivial families of functions. It proves that, under suitable coherence assumptions, enriched rho-functions derived from $n$-dimensional walks (notably $\rho_2^n$ and the enriched $\phncfamily\textphnc{r}_n$) display coherence and, in important cases, nontriviality, linking these phenomena to Čech cohomology on ordinals with the order topology. The authors develop forcing to add higher-square sequences and establish consistency results about when such principles hold or fail, including the constructible universe and large-cardinal contexts. They demonstrate nontriviality results for $n$-dimensional walks, show how truncated and simulated walks connect different dimensions, and discuss obstacles to obtaining nontrivial coherent $n$-dimensional families purely from higher walks for $n>1$, highlighting subtle interactions with set-theoretic forcing and combinatorial topology. Overall, the work provides a framework connecting higher-dimensional combinatorics on cardinals, forcing, and cohomological invariants, deepening our understanding of incompactness phenomena at and above $\omega_n$.

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 $λ$ entail certain non-vanishing Čech cohomology groups for $λ$ considered with the order topology.

Higher walks and squares

TL;DR

The paper advances higher-dimensional analogues of Todorcevic's walks on ordinals by introducing coherent --sequences and higher square principles, and by defining enriched higher-walk constructs that yield coherent yet nontrivial families of functions. It proves that, under suitable coherence assumptions, enriched rho-functions derived from -dimensional walks (notably and the enriched ) display coherence and, in important cases, nontriviality, linking these phenomena to Čech cohomology on ordinals with the order topology. The authors develop forcing to add higher-square sequences and establish consistency results about when such principles hold or fail, including the constructible universe and large-cardinal contexts. They demonstrate nontriviality results for -dimensional walks, show how truncated and simulated walks connect different dimensions, and discuss obstacles to obtaining nontrivial coherent -dimensional families purely from higher walks for , highlighting subtle interactions with set-theoretic forcing and combinatorial topology. Overall, the work provides a framework connecting higher-dimensional combinatorics on cardinals, forcing, and cohomological invariants, deepening our understanding of incompactness phenomena at and above .

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 -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 -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 entail certain non-vanishing Čech cohomology groups for considered with the order topology.

Paper Structure

This paper contains 11 sections, 36 theorems, 75 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Preliminaries on coherence and triviality
  4. Higher squares
  5. Forcing to add higher square sequences
  6. Higher walks
  7. Preliminaries
  8. The case $n=3$
  9. Interlude: truncated and simulated walks
  10. The general case
  11. Nontriviality

Key Result

Proposition 3.3

The sequence $\mathcal{C}^\delta = \langle C^\delta_{\vec{\gamma}} \mid \vec{\gamma} \in I(\mathcal{C}^\delta) \rangle$ is an $n$-$C$-sequence on $C_\delta$. Moreover, if $\mathcal{C}$ is $(n+1)$-coherent, then $\mathcal{C}^\delta$ is $n$-coherent.

Figures (5)

  • Figure 3:
  • Figure 4: The ? indicates that we do not know whether the node $(\xi,\eta_\xi,\gamma^*,\delta^*)$ is terminal. Compare with Figure \ref{['first_step_bad']}.
  • Figure 9: The bad case. By Lemma \ref{['badNodes']}, the parts of $S_3(\alpha,\beta^*,\gamma^*,\delta^*)$ and $S_3(\xi,\beta^*,\gamma^*,\delta^*)$ that lie within the dashed lines are the same.
  • Figure 10: The good case. We have assumed for definiteness that the starting node is $(\alpha,\beta,\gamma,\delta)$. Here, the left and the right trees are the same, by Lemma \ref{['7.7']}.
  • Figure 15: The effect of stretching a full $2$-tree into a full $4$-tree.

Theorems & Definitions (96)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • Proposition 3.4
  • proof
  • Definition 3.5
  • Proposition 3.6
  • proof
  • ...and 86 more