arXiv subjects

Torgeir Aambø

Publications and source records attributed to Torgeir Aambø.

7 recordsLinked to original sources

arXiv2026

Possibilistic operators in Formal Concept Analysis as Kan extensions

In this paper we prove that Dubois--Prade's eight possibilistic operators in Formal Concept Analysis arise canonically from Kan extensions of the underlying boolean profunctor. This provides a conceptual explanation for the result that $NΠ$-pairs are the formal concepts of the complement context. We further prove that the FCA closure operator and the $NΠ$-pairs are the only symmetric or asymmetric operator compositions that give formal concepts. Finally we use these eight possibilistic operators to construct new closure operators on a formal context via standard categorical arguments.

math.CT↗

arXiv2026

Collective deterrence as a classification problem: Voting rules, deterrence credibility, and escalation risk

Deterrence coalitions that collectively own their deterrence technology, need an institutional design to decide when to retaliate against an attack or incident. This choice of institutional design, formalized through a social choice function, introduces a tradeoff between credible deterrence and escalation risk. We study this tradeoff via a simple signalling model, and use it to construct an associated binary classification problem to determine institutional designs that perform well in a variety of environments. For a small coalition of four members, we compute and study the statistics of the empirical ROC curves associated to a variety of choice functions and probability distributions for retaliation and false positives.

math.OC↗

arXiv2026

A categorical formalization of epistemic uncertainty frameworks

Epistemic uncertainty arises in lack of complete knowledge about the state of a system. There are multiple mathematical frameworks for measuring such uncertainty quantitatively, often referred to as imprecise probability theories. Inspired by work of Opdan, we introduce a general category theoretic definition of epistemic calculi, which we use as a foundation for modelling and studying contradictions and synergies between several philosophical epistemological concepts. We further develop an enriched category theoretic process for changing calculi, and use this to study relationships between existing examples, like possibility theory and certainty factors. Finally, we introduce a general categorical form of belief updating based on change of enrichment, and prove that Bayesian updating and possibilistic conditioning arise as examples.

math.CT↗

arXiv2025

Positselski duality in $\infty$-categories

We introduce the notion of a contramodule over a cocommutative coalgebra in a presentably symmetric monoidal $\infty$-category $\mathcal{C}$, and prove a symmetric monoidal $\infty$-categorical version of Positselski's comodule-contramodule correspondence when the coalgebra is coidempotent. This gives a new perspective on, and a new proof of local duality -- in the sense of Hovey--Palmieri--Strickland and Dwyer--Greenlees -- whenever $\mathcal{C}$ is stable and compactly generated. We further consider an analog of Positselski's definition of contramodules over topological rings in the $\infty$-categorical setting, and show that the two perspectives on contramodules are equivalent. As examples we describe the categories of $K(n)$-local spectra, $T(n)$-local spectra and the derived complete category of a ring $R$, as categories of contramodules.

math.AT↗

arXiv2024

Classification of localizing subcategories along t-structures

We study the interplay between localizing subcategories in a stable $\infty$-category $\mathcal{C}$ with $t$-structure $(\mathcal{C}_{\geq 0}, \mathcal{C}_{\leq 0})$, the prestable $\infty$-category $\mathcal{C}_{\geq 0}$ and the abelian category $\mathcal{C}^{\heartsuit}$. We prove that weak localizing subcategories of $\mathcal{C}^{\heartsuit}$ are in bijection with the localizing subcategories of $\mathcal{C}$ where object-containment can be checked on the heart. This generalizes similar known correspondences for noetherian rings and bounded $t$-structures. We also prove that this restricts to a bijection between localizing subcategories of $\mathcal{C}^{\heartsuit}$, and localizing subcategories of $\mathcal{C}$ that are kernels of $t$-exact functors -- lifting Lurie's correspondence between localizing subcategories in $\mathcal{C}_{\geq 0}$ and $\mathcal{C}^{\heartsuit}$ to the stable category $\mathcal{C}$.

math.AT↗

arXiv2024

Algebraicity in monochromatic homotopy theory

Using Patchkoria--Pstrągowski's version of Franke's algebraicity theorem, we prove that the category of $K_p(n)$-local spectra is exotically equivalent to the category of derived $I_n$-complete periodic comodules over the Adams Hopf algebroid $(E_*, E_*E)$ for large primes. This gives a finite prime result analogous to the asymptotic algebraicity for $\mathrm{Sp}_{K_p(n)}$ of Barthel--Schlank--Stapleton.

math.AT↗

arXiv2021

Formality of spaces with Lusternik-Schnirelmann category 1

It is a well known fact that formal dg-algebras admit no non-trivial Massey products, while the converse fails. We prove that by restricting to dg-algebras whose induced product on cohomology is trivial, we do in fact get this converse. This allows us to prove that spaces of Lusternik-Schnirelmann category 1 are formal spaces.

math.AT↗