Duality in tensor-triangular geometry via proxy-smallness

Thomas Peirce; Jordan Williamson

Home/Archive/arXiv

arXiv · 2510.24415

Duality in tensor-triangular geometry via proxy-smallness

Thomas Peirce·Jordan Williamson

View original source ↗Read source document ↗Export metadata ↓

Abstract

We make a systematic study of duality phenomena in tensor-triangular geometry, generalising and complementing previous results of Balmer--Dell'Ambrogio--Sanders and Dwyer--Greenlees--Iyengar. A key feature of our approach is the use of proxy-smallness to remove assumptions on functors preserving compact objects, and to this end we introduce proxy-small geometric functors and establish their key properties. Given such a functor, we classify the rigid objects in its associated torsion category, giving a new perspective on results of Benson--Iyengar--Krause--Pevtsova. As a consequence, we show that any proxy-small geometric functor satisfies Grothendieck duality on a canonical subcategory of objects, irrespective of whether its right adjoint preserves compact objects. We use this as a tool to classify Matlis dualising objects and to provide a suitable generalisation of the Gorenstein ring spectra of Dwyer--Greenlees--Iyengar in tensor-triangular geometry. We illustrate the framework developed with various examples and applications, showing that it captures Matlis duality and Gorenstein duality in commutative algebra, duality phenomena in chromatic and equivariant stable homotopy theory, and Watanabe's theorem in polynomial invariant theory.

Explore related subjects

math.CT math.AC math.AT math.RT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Thomas Peirce, Jordan Williamson. 2026-05-25. Duality in tensor-triangular geometry via proxy-smallness. https://arxiv.org/abs/2510.24415

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Categorical Generalization of Counterpoint

We extend Mazzola's counterpoint model using category theory, generalizing from the category $\mathbf{Set}$ to an arbitrary topos other than $\mathbf{Set}$. This generalization suggests that counterpoint's essential structure depends on specific categorical conditions rather than classical set-theoretic reasoning. A key contribution is identifying sufficient requirements for a well-behaved counterpoint theory in a topos: some version of Zorn's Lemma (GJZL), and two-valuedness and split supports (NS). Within a topos, we introduce (weak) quasidichotomies alongside the classical notion of dichotomy. These structures capture varying degrees of oppositional structure between consonance and dissonance, with weak quasidichotomies preserving the non-Boolean flexibility essential to musical practice while quasidichotomies represent maximal opposition short of complete partition. We prove a generalized counterpoint theorem giving sufficient conditions for the existence of admitted successors. When the ambient topos turns non-zero successor objects into points, admitted succession can be iterated to form counterpoint paths, which may terminate at consonances with no admitted successor. The framework naturally accommodates counterpoint with sets instead of pure pitches, relaxing the ``yes/no'' character of classical consonance definitions and emphasizing context-dependence. Mazzola's model allows a Kuratowski closure operator induced by a polarity, which defines an internal topology enabling algebraic-topological analysis of counterpoint structure. We conclude by showing this construction generalizes to involutive morphisms. This categorical approach provides foundations for understanding both the historical evolution of contrapuntal practice and cross-cultural divergences in interval organization.

math.CT↗

arXiv2026

From 3-crossed modules to Gray-type 4-categories

In this paper, we investigate the relation between the category of 3-crossed modules and the category of Gray-type 4-groups. The notion of a 3-crossed module was first introduced by Arvasi \textit{et al.}, motivated by the question of what kind of algebraic structure completely encodes a homotopy 4-type. On the other hand, from the point of view that higher groups are equivalent to algebraic realizations of higher categories -- as exemplified by the relationship between 2-crossed modules and Gray 3-groups established by Sarikaya--Ulualan -- it had not been clear how the 3-crossed modules of Arvasi \textit{et al.} relate to any higher category. In our previous paper, we proposed a new definition of a 3-crossed module and observed that it admits a natural interpretation in terms of higher categories. In this paper, we make this interpretation precise: we introduce a 4-category, which reduces to a semistrict braided monoidal 2-category when restricted to a single object and a single 1-morphism, and prove that the category of our 3-crossed modules is equivalent to the category of Gray 4-groups, defined as single-object versions of this 4-category in which all morphisms are invertible. We therefore expect that these structures can correctly capture the topological nature of surface knots and higher-dimensional manifolds.

math.CT↗

arXiv2026

Observations on the variety of equationally linear Heyting semilattices

In previous work, we analysed a number of categorical properties, of interest in the context of Janelidze-Márki-Tholen semi-abelian categories, for the variety $\mathsf{HSLat}$ of Heyting semilattices. In this paper, we focus on the subvariety $\mathsf{ELHSLat}$ of equationally linear Heyting semilattices. Our main objective is to show that, unlike $\mathsf{HSLat}$, this category is algebraically coherent. We furthermore prove that it is neither locally algebraically cartesian closed nor cosmash associative.

math.CT↗