arXiv subjects

Nathaniel Bannister

Publications and source records attributed to Nathaniel Bannister.

8 recordsLinked to original sources

arXiv2026

Merging $\lim^1 \mathbf{A} \ne 0$ with other nonvanishing constructions

We develop methods for forcing $\lim^1 \mathbf{A} \ne 0$, where $\mathbf{A}$ is a particular inverse system of abelian groups introduced by Marde\v{s}i\'c and Prasolov in their computation of certain strong homology groups. These methods allow us to extend previous nonvanishing results of Casarosa and Lambie-Hanson for $\lim^k \mathbf{A}$ for $k \geq 2$. Specifically we show that, for a given $n$, it is relatively consistent with ZFC that $\mathfrak{b} = \mathfrak{d} = \omega_n$ and $\lim^k \mathbf{A} \ne 0$ whenever $1 \leq k \leq n$ (previously established with $2 \leq k \leq n$). We also show it is relatively consistent with ZFC that $\mathfrak{b} = \mathfrak{d} = \omega_{\omega+2}$ and $\lim^k \mathbf{A} \ne 0$ for all $k \geq 1$ (previously established with $k \geq 2$). We also adapt proofs of Kamo to show that $\lim^1 \mathbf{A} = 0$ holds in many finite support iterated forcing extensions.

math.LO↗

arXiv2026

Condensed Sets and the Solovay Model

We exhibit a geometric morphism from the Grothendieck topos representing the Solovay model to the $\kappa$-pyknotic sets of Barwick--Haine and Clausen--Scholze. We then use the properties of this morphism and automatic continuity in the Solovay model to prove Clausen--Scholze's resolution of the Whitehead problem for discrete condensed abelian groups. We also exhibit an analogous internal $Ext$ computation between locally compact abelian groups in the Solovay model.

math.LO↗

arXiv2025

Nonvanishing Higher Derived Limits without $w\diamondsuit_{\omega_1}$

We prove a common refinement of theorems of Bergfalk and of Casarosa and Lambie-Hanson, showing that under certain hypotheses, the higher derived limits of a certain inverse system of abelian groups $\mathbf{A}$ do not vanish. The refined theorem has a number of interesting corollaries, including the nonvanishing of the second derived limit of $\mathbf{A}$ in many of the common models of set theory of the reals and in the Mitchell model. In particular, we disprove a conjecture of Bergfalk, Hru\v{s}\'ak, and Lambie-Hanson that higher derived limits of $\mathbf{A}$ vanish in the Miller model.

math.LO↗

arXiv2025

All you need is $\mathbf{A}_\kappa$

We show that the vanishing of higher derived limits of the system $\mathbf{A}_\kappa$ implies the additivity of strong homology on the class of locally compact metric spaces of weight at most $\kappa$, thereby establishing a converse to a theorem of Marde\v{s}i\'c and Prasolov.

math.LO↗

arXiv2024

Locales as spaces in outer models

Let M be a transitive model of set theory and X be a space in the sense of M. Is there a reasonable way to interpret X as a space in V? A general theory due to Zapletal provides a natural candidate which behaves well on sufficiently complete spaces (for instance \v{C}ech complete spaces) but behaves poorly on more general spaces - for instance, the Zapletal interpretation does not commute with products. We extend Zapletal's framework to instead interpret locales, a generalization of topological spaces which focuses on the structure of open sets. Our extension has a number of desirable properties; for instance, localic products always interpret as spatial products. We show that a number of localic notions coincide exactly with properties of their interpretations; for instance, we show a locale is $T_U$ if and only if all its interpretations are $T_1$, a locale is $I$-Hausdorff if and only if all its interpretations are $T_2$, a locale is regular if and only if all its interpretations are $T_3$, and a locale is compact if and only if all its interpretations are compact.

math.LO↗

arXiv2023

A descriptive approach to higher derived limits

We present a new aspect of the study of higher derived limits. More precisely, we introduce a complexity measure for the elements of higher derived limits over the directed set $Ω$ of functions from $\mathbb{N}$ to $\mathbb{N}$ and prove that cocycles of this complexity are images of cochains of the roughly the same complexity. In the course of this work, we isolate a partition principle for powers of directed sets and show that whenever this principle holds, the corresponding derived limit $\mathrm{lim}^n$ is additive; vanishing results for this limit are the typical corollary. The formulation of this partition hypothesis synthesizes and clarifies several recent advances in this area.

math.LO↗

arXiv2023

Additivity of derived limits in the Cohen model

We show that, in the model constructed by adding sufficiently many Cohen reals, derived limits are additive on a large class of systems. This generalizes the work of Jeffrey Bergfalk, Michael Hru\v s\'ak, and Chris Lambie-Hanson which focuses on the system $\mathbf{A}$. In the process, we isolate a partition principle responsible for the vanishing of derived limits on collections of Cohen reals and reframe the propagating trivializations results of Bergfalk, Hru\v s\'ak and Lambie-Hanson as a theorem of ZFC. In light of results of the author, Jeffrey Bergfalk, and Justin Moore, the additivity of derived limits also implies additivity results for strong homology.

math.LO↗

arXiv2022

On the additivity of strong homology for locally compact separable metric spaces

We show that it is consistent relative to a weakly compact cardinal that strong homology is additive and compactly supported within the class of locally compact separable metric spaces. This complements work of Mardešić and Prasolov showing that the Continuum Hypothesis implies that a countable sum of Hawaiian earrings witnesses the failure of strong homology to possess either of these properties. Our results build directly on work of Lambie-Hanson and the second author which establishes the consistency, relative to a weakly compact cardinal, of $\mathrm{lim}^s \mathbf{A} = 0$ for all $s \geq 1$ for a certain pro-abelian group $\mathbf{A}$; we show that that work's arguments carry implications for the vanishing and additivity of the $\mathrm{lim}^s$ functors over a substantially more general class of pro-abelian groups indexed by $\mathbb{N}^{\mathbb{N}}$.

math.LO↗