arXiv subjects

Marius Thaule

Publications and source records attributed to Marius Thaule.

6 recordsLinked to original sources

arXiv2025

Lifting Cocycles: From Heuristic to Theory

The circular coordinates algorithm, a key tool in topological data analysis, relies on a theoretically unvalidated lifting step to convert cocycles from a prime field to integer coefficients. We provide a rigorous analysis of this procedure, establishing a criterion for its success. We also introduce a novel algebraic method to reduce any lifted cocycle to a cocycle with winding number 1, ensuring feature correctness. These principles are extended to homology cycles, solidifying the theoretical foundation of this widely used feature extraction technique.

math.AT↗

arXiv2023

Toda brackets in n-angulated categories

We introduce Toda brackets for n-angulated categories and show that the various definitions of Toda brackets coincide. We prove juggling formulas for these Toda brackets generalizing the triangulated case. Following that, we generalize a theorem due to Heller in the triangulated setting to the setting of n-angulated categories. We also provide several examples of computing Toda brackets for n-angulated categories. Finally, for an n-angulated category sitting in a triangulated category as in the setup of Geiss, Keller and Oppermann, we show that Toda brackets in the n-angulated sense coincide with n-fold Toda brackets in the triangulated sense up to an explicit sign.

math.CT↗

arXiv2016

The morphism axiom for n-angulated categories

The morphism axiom for n-angulated categories states that a morphism between the bases of two n-angles can be extended to a morphism of n-angles. We show that this axiom is redundant. For triangulated categories, this was proved by J.P. May.

math.CT↗

arXiv2015

Higher n-angulations from local rings

We show that the category of finitely generated free modules over certain local rings is n-angulated for every n at least 3. In fact, we construct several classes of n-angles, parametrized by equivalence classes of units in the local rings. Finally, we show that for odd values of n some of these n-angulated categories are not algebraic.

math.CT↗

arXiv2012

The Grothendieck group of an n-angulated category

We define the Grothendieck group of an n-angulated category and show that for odd n its properties are as in the special case of n=3, i.e. the triangulated case. In particular, its subgroups classify the dense and complete n-angulated subcategories via a bijective correspondence. For a tensor n-angulated category, the Grothendieck group becomes a ring, whose ideals classify the dense and complete n-angulated tensor ideals of the category.

math.CT↗

arXiv2012

The axioms for n-angulated categories

We discuss the axioms for an n-angulated category, recently introduced by Geiss, Keller and Oppermann. In particular, we introduce a higher octahedral axiom, and show that it is equivalent to the mapping cone axiom for an n-angulated category. For a triangulated category, the mapping cone axiom, our octahedral axiom and the classical octahedral axiom are all equivalent.

math.CT↗