arXiv subjects

Souheila Hassoun

Publications and source records attributed to Souheila Hassoun.

8 recordsLinked to original sources

arXiv2025

Stratifying systems and Jordan-Hölder extriangulated categories

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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arXiv2023

On the lattices of exact and weakly exact structures

We initiate in this article the study of weakly exact structures, a generalization of Quillen exact structures. We introduce weak counterparts of one-sided exact structures and show that a left and a right weakly exact structure generate a weakly exact structure. We further define weakly extriangulated structures on an additive category $\mathcal{A}$ and characterize weakly exact structures among them. We investigate when these structures on $\mathcal{A}$ form lattices. We prove that the lattice of substructures of a weakly extriangulated structure is isomorphic to the lattice of topologizing subcategories of a certain functor category. In the idempotent complete case, we characterize the lattice of all weakly exact structures and we prove the existence of a unique maximal weakly exact structure. We study in detail the situation when $\mathcal{A}$ is additively finite, giving a module-theoretic characterization of closed sub-bifunctors of $\mbox{Ext}^1$ among all additive sub-bifunctors.

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arXiv2021

Double framed moduli spaces of quiver representations

Motivated by problems in the neural networks setting, we study moduli spaces of double framed quiver representations and give both a linear algebra description and a representation theoretic description of these moduli spaces. We define a network category whose isomorphism classes of objects correspond to the orbits of quiver representations, in which neural networks map input data. We then prove that the output of a neural network depends only on the corresponding point in the moduli space. Finally, we present a different perspective on mapping neural networks with a specific activation function, called ReLU, to a moduli space using the symplectic reduction approach to quiver moduli.

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arXiv2021

Reduction of exact structures

Examples of exact categories in representation theory are given by the category of Delta-filtered modules over quasi-hereditary algebras, but also by various categories related to matrix problems, such as poset representations or representations of bocses. Motivated by the matrix problem background, we study in this article the reduction of exact structures, and consider the poset Ex(A) of all exact structures on a fixed additive category A. This poset turns out to be a complete lattice, and under suitable conditions results of Enomoto's imply that it is boolean. We initiate in this article a detailed study of exact structures E by generalizing notions from abelian categories such as the length of an object relative to E and the quiver of an exact category (A,E). We investigate the Gabriel-Roiter measure for (A,E), and further study how these notions change when the exact structure varies.

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arXiv2021

Intersections, sums, and the Jordan-Hölder property for exact categories

We investigate how the concepts of intersection and sums of subobjects carry to exact categories. We obtain a new characterisation of quasi-abelian categories in terms of admitting admissible intersections in the sense of Hassoun and Roy. There are also many alternative characterisations of abelian categories as those that additionally admit admissible sums and in terms of properties of admissible morphisms. We then define a generalised notion of intersection and sum which every exact category admits. Using these new notions, we define and study classes of exact categories that satisfy the Jordan-Hölder property for exact categories, namely the Diamond exact categories and Artin-Wedderburn exact categories. By explicitly describing all exact structures on $\mathcal{A}= \mbox{rep}\, Λ$ for a Nakayama algebra $Λ$ we characterise all Artin-Wedderburn exact structures on $\mathcal{A}$ and show that these are precisely the exact structures with the Jordan-Hölder property.

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arXiv2020

Examples and non-examples of integral categories and the admissible intersection property

Integral categories form a sub-class of pre-abelian categories whose systematic study was initiated by Rump in 2001. In the first part of this article we determine whether several categories of topological and bornological vector spaces are integral. Moreover, we establish that the class of integral categories is not contained in the class of quasi-abelian categories, and that there exist semi-abelian categories that are neither integral nor quasi-abelian. In the last part of the article we show that a category is quasi-abelian if and only if it has admissible intersections, in the sense considered recently by Br{ü}stle, Hassoun and Tattar. This exhibits that a rich class of non-abelian categories having this property arises naturally in functional analysis.

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arXiv2020

Admissible intersection and sum property

We introduce subclasses of exact categories in terms of admissible intersections or admissible sums or both at the same time. These categories are recently studied by Brüstle, Hassoun, Shah, Tattar and Wegner to give characterisations of quasi-abelian and abelian categories. We also generalise the Schur lemma to the context of exact categories.

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arXiv2020

Integral and quasi-abelian hearts of twin cotorsion pairs on extriangulated categories

It was shown recently that the heart of a twin cotorsion pair ((S,T),(U,V)) on an extriangulated category is semi-abelian. We provide a sufficient condition for the heart to be integral and another for the heart to be quasi-abelian. This unifies and improves the corresponding results for exact and triangulated categories. Furthermore, if T=U, then we show that the Gabriel-Zisman localisation of the heart at the class of its regular morphisms is equivalent to the heart of the single twin cotorsion pair (S,T). This generalises and improves the known result for triangulated categories, thereby providing new insights in the exact setting.

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