Stratifying Systems and the Jordan-Hölder Property in Extriangulated Categories

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This research paper introduces the concept of stratifying systems in extriangulated categories to establish the Jordan-Hölder property, demonstrating that certain subcategories of extriangulated categories inherit a structure where composition series-like filtrations have well-defined lengths and "composition factors".

Samenvatting

• Bibliographic Information: Brüstle, T., Hassoun, S., Shah, A., & Tattar, A. (2024). Stratifying systems and Jordan-Hölder extriangulated categories. arXiv preprint arXiv:2208.07808v4.

• Research Objective: This paper aims to extend the study of length and the Jordan-Hölder property to the realm of extriangulated categories, building upon previous work in module, triangulated, and exact categories. The authors introduce the concept of stratifying systems in this context to investigate the Jordan-Hölder property for certain subcategories.

• Methodology: The authors develop the theory of stratifying systems within the framework of extriangulated categories. They define E-stratifying systems and projective E-stratifying systems, drawing parallels to similar concepts in other categorical settings. The paper utilizes the properties of extriangles, weak idempotent completeness, and the Grothendieck monoid to establish its results.

• Key Findings:

  • Every stratifying system in an extriangulated category can be embedded into a minimal projective one.
  • Under specific conditions, the subcategory of objects filtered by a projective E-stratifying system forms a length, Jordan-Hölder extriangulated category.
  • The authors characterize length Jordan-Hölder extriangulated categories via properties of their associated Grothendieck monoid and Grothendieck group.
  • The paper provides a counterexample to a recent question posed by Enomoto–Saito, showing that a skeletally small extriangulated category with a reduced Grothendieck monoid need not be exact.

• Main Conclusions: The research demonstrates that the well-behaved notion of length and the Jordan-Hölder property, typically studied in the context of abelian and exact categories, can be extended to the more general setting of extriangulated categories. This opens up new avenues for studying structural properties of these categories.

• Significance: This work contributes significantly to the understanding of extriangulated categories, a relatively new area of study in category theory. By establishing the Jordan-Hölder property in this context, the authors provide valuable tools and insights for further research in this field and related areas.

• Limitations and Future Research: The paper primarily focuses on the theoretical aspects of stratifying systems and the Jordan-Hölder property in extriangulated categories. Further research could explore specific examples and applications of these concepts in areas like representation theory, where extriangulated categories have shown promise. Additionally, investigating the behavior of these properties under various categorical constructions could be a fruitful direction for future work.

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Stratifying systems and Jordan-H\"{o}lder extriangulated categories

by Thom... om arxiv.org 10-21-2024

https://arxiv.org/pdf/2208.07808.pdf

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How do the concepts of stratifying systems and the Jordan-Hölder property in extriangulated categories inform the study of specific algebraic structures, such as certain classes of rings or algebras?

Answer:
Stratifying systems and the Jordan-Hölder property in extriangulated categories provide powerful tools for studying specific algebraic structures by offering a framework to decompose and analyze their representation categories. Here's how:

Identifying well-behaved subcategories: Stratifying systems, particularly projective E-stratifying systems, identify subcategories within the representation category of an algebra that possess desirable properties like the Jordan-Hölder property. This means objects in these subcategories admit unique decompositions (up to isomorphism and permutation) into "simpler" objects, mirroring the decomposition of modules over a quasi-hereditary algebra into standard modules.

Generalizing classical notions: The framework extends classical notions of composition series and the Jordan-Hölder theorem from the realm of modules over rings to more abstract settings like extriangulated categories. This allows us to study a broader class of algebras whose representation categories might not be module categories but still exhibit similar decomposition behavior.

Connecting to structural properties: The existence of stratifying systems and the Jordan-Hölder property in the representation category can often be linked back to specific structural properties of the algebra itself. For example, the presence of a stratifying system might indicate the algebra is standardly stratified or that it can be studied using techniques from highest weight categories.
Specific examples:

Quasi-hereditary algebras: The original motivation for stratifying systems stemmed from studying quasi-hereditary algebras. Their representation categories admit stratifying systems, and the standard modules form the "simple" objects in the corresponding Jordan-Hölder subcategory.

Stratified algebras: More generally, the concept of stratified algebras, which generalize quasi-hereditary algebras, is closely tied to the existence of stratifying systems in their representation categories.

Other examples: The framework is applicable beyond just finite-dimensional algebras. It can be used to study representations of Lie algebras, algebraic groups, and even categories appearing in other areas of mathematics like algebraic geometry.
In summary, stratifying systems and the Jordan-Hölder property provide a lens through which we can analyze the representation theory of various algebraic structures. By understanding how objects decompose in these categories, we gain insights into the structure of the algebras themselves and can leverage these decompositions to study their representations more effectively.

Could there be alternative conditions on extriangulated categories or stratifying systems that lead to different or weaker forms of the Jordan-Hölder property?

Answer:
Yes, it's certainly possible to explore alternative conditions on extriangulated categories or stratifying systems that could lead to different or weaker forms of the Jordan-Hölder property. Here are some potential avenues for investigation:
Weaker forms of the Jordan-Hölder property:

Relaxing uniqueness: Instead of requiring strict equivalence of composition series, we could consider weaker notions of equivalence. For example, we might only require the lengths of any two composition series to be equal, or that the multisets of isomorphism classes of factors coincide.

Local Jordan-Hölder property: We could investigate conditions under which the Jordan-Hölder property holds only for a specific class of objects within the extriangulated category, rather than for all objects. This would be akin to studying modules of finite length within a larger module category.
Alternative conditions on extriangulated categories:

Variations on length: Instead of requiring the existence of finite composition series, we could explore notions of "length" based on transfinite induction or other ordinal-based approaches. This might be relevant for studying categories with objects that admit infinite filtrations.

Axiomatic approaches: One could explore modifications to the axioms of extriangulated categories themselves to guarantee weaker forms of the Jordan-Hölder property. For instance, relaxing the "additivity" of the realization functor might lead to interesting variations.
Alternative conditions on stratifying systems:

Weakening orthogonality: The orthogonality conditions imposed on stratifying systems could be relaxed. This might lead to subcategories with a weaker form of the Jordan-Hölder property, where the decomposition into "simple" objects is not as tightly controlled.

Generalized projective objects: The notion of "projective" objects in a stratifying system could be generalized. This might involve considering different notions of projectivity or exploring relative homological algebra within the extriangulated category.
Exploring these alternative conditions could lead to a richer understanding of decomposition theory in extriangulated categories. It might uncover new classes of categories with interesting decomposition behavior and provide valuable insights into the interplay between the structure of an extriangulated category and the Jordan-Hölder property.

If we view the Jordan-Hölder property as a form of "decomposition theory," what other mathematical objects or structures might admit similar decompositions with desirable properties?

Answer:
The Jordan-Hölder property, viewed as a form of "decomposition theory," has analogues in various mathematical contexts beyond extriangulated categories. The common thread is the ability to decompose objects into "simpler" components in a meaningful and often unique way. Here are some examples:
Algebra:

Groups: Finite groups admit a decomposition into simple groups via their composition series. This decomposition is unique up to permutation and isomorphism, analogous to the Jordan-Hölder property.

Modules over rings: Modules over a ring can often be decomposed into simpler modules, such as indecomposable modules or cyclic modules. While not always unique, these decompositions provide valuable information about the module's structure.
Topology:

CW complexes: In algebraic topology, CW complexes are built by attaching cells of increasing dimension. This decomposition into cells provides a powerful tool for studying the homotopy and homology of spaces.

Manifolds: Manifolds can be decomposed into charts, which are open sets homeomorphic to Euclidean space. This decomposition is crucial for developing calculus and analysis on manifolds.
Combinatorics:

Matroids: Matroids abstract the notion of independence found in linear algebra and graph theory. They admit decompositions into "simpler" matroids, such as circuits, flats, and bases, which provide insights into their structure.

Graphs: Graphs can be decomposed into connected components, cycles, trees, and other subgraphs. These decompositions are fundamental for studying graph properties and algorithms.
Other areas:

Representations of Lie algebras and algebraic groups: Similar to the case of modules over rings, representations of Lie algebras and algebraic groups can often be decomposed into simpler representations, such as irreducible representations or highest weight modules.

Sheaves: In algebraic geometry, sheaves are mathematical objects that "glue" local data together. Sheaves can be decomposed into simpler sheaves, such as skyscraper sheaves and locally free sheaves, which provide insights into the geometry of the underlying space.
The desirable properties of these decompositions often include:

Uniqueness: The decomposition should be unique in some sense, either up to isomorphism and permutation or with respect to some other equivalence relation.

Finiteness: The decomposition should involve a finite number of components, at least in well-behaved cases.

Meaningful components: The components of the decomposition should be "simpler" in some sense and provide useful information about the original object.
The search for and study of such decomposition theories is a recurring theme in mathematics. It reflects a desire to understand complex objects by breaking them down into more manageable pieces and analyzing how these pieces fit together.