Stratifying systems and Jordan-Hölder extriangulated categories Article Swipe

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Thomas Brüstle , Souheila Hassoun , Amit Shah , Aran Tattar ·

YOU? · · 2022 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2208.07808

Stratifying systems, which have been defined for module, triangulated and exact categories previously, were developed to produce examples of standardly stratified algebras. A stratifying system $Φ$ is a finite set of objects satisfying some orthogonality conditions. One very interesting property is that the subcategory $\mathcal{F}(Φ)$ of objects admitting a composition series-like filtration with factors in $Φ$ has the Jordan-Hölder property on these filtrations. This article has two main aims. First, we introduce notions of subobjects, simple objects and composition series for an extriangulated category, in order to define a Jordan-Hölder extriangulated category. Moreover, we characterise Jordan-Hölder, length, weakly idempotent complete extriangulated categories in terms of the associated Grothendieck monoid and Grothendieck group. Second, we develop a theory of stratifying systems in extriangulated categories. We define projective stratifying systems and show that every stratifying system $Φ$ in an extriangulated category is part of a minimal projective one $(Φ,Q)$. We prove that $\mathcal{F}(Φ)$ is a length, Jordan-Hölder extriangulated category when $(Φ,Q)$ satisfies a left exactness condition. We give several examples and answer a recent question of Enomoto--Saito in the negative.

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Mathematics Subcategory Idempotence Grothendieck group Orthogonality Pure mathematics Series (stratigraphy) Property (philosophy) Combinatorics Composition (language) 2-category Enriched category Algebra over a field Linguistics Functor Geometry Paleontology Philosophy Epistemology Biology Abelian group

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2208.07808
PDF: https://arxiv.org/pdf/2208.07808
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Cited By: 2
Related Works: 10
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DOI: https://doi.org/10.48550/arxiv.2208.07808

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Title: Stratifying systems and Jordan-Hölder extriangulated categories

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Language: en

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Publication year: 2022

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Publication date: 2022-08-16

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Authors: Thomas Brüstle, Souheila Hassoun, Amit Shah, Aran Tattar

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Landing page: https://arxiv.org/abs/2208.07808

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Concepts: Mathematics, Subcategory, Idempotence, Grothendieck group, Orthogonality, Pure mathematics, Series (stratigraphy), Property (philosophy), Combinatorics, Composition (language), 2-category, Enriched category, Algebra over a field, Linguistics, Functor, Geometry, Paleontology, Philosophy, Epistemology, Biology, Abelian group

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Cited by: 2

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abstract_inverted_index.Grothendieck	107, 110
abstract_inverted_index.characterise	94
abstract_inverted_index.filtrations.	62
abstract_inverted_index.triangulated	8
abstract_inverted_index.orthogonality	34
abstract_inverted_index.Enomoto--Saito	174
abstract_inverted_index.Jordan-Hölder	58, 89, 154
abstract_inverted_index.extriangulated	82, 90, 100, 121, 137, 155
abstract_inverted_index.Jordan-Hölder,	95
abstract_inverted_index.$\mathcal{F}(Φ)$	44, 150
cited_by_percentile_year
countries_distinct_count	0
institutions_distinct_count	4
sustainable_development_goals[0].id	https://metadata.un.org/sdg/10
sustainable_development_goals[0].score	0.4099999964237213
sustainable_development_goals[0].display_name	Reduced inequalities
citation_normalized_percentile