The Exact Structure of Triangulated Categories: Unveiling Abelian Foundations
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2025
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| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | Triangulated categories are foundational structures in algebraic geometry, representation theory, and homological algebra. While their axiomatic definition provides a powerful framework for studying objects up to quasi-isomorphism, their underlying ``exact structure'' often remains implicit or is primarily understood through the lens of derived functors from abelian categories. This paper rigorously delves into the precise nature of this exact structure, proposing and demonstrating methodologies to explicitly unveil the abelian foundations that frequently give rise to these triangulated categories. We systematically investigate the relationship between exact categories, Frobenius categories, and their associated stable categories, demonstrating how the latter, often being triangulated, retain crucial information about the former. Special attention is given to the construction of exact sequences and distinguished triangles, exploring how the homological properties encoded within an abelian category manifest in its derived or stable category. Through rigorous theoretical analysis and constructive proofs, we provide a clearer understanding of how the intricate geometry and algebra of triangulated categories are ultimately rooted in more conventional abelian structures, offering new perspectives for their classification and manipulation. Our work makes explicit the processes by which abelian exactness translates into triangulated axioms and shows how abelian ``hearts'' can be extracted from complex triangulated structures. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17809014 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Exact Structure of Triangulated Categories: Unveiling Abelian Foundations Revista, Zen MATH, 10 Triangulated categories are foundational structures in algebraic geometry, representation theory, and homological algebra. While their axiomatic definition provides a powerful framework for studying objects up to quasi-isomorphism, their underlying ``exact structure'' often remains implicit or is primarily understood through the lens of derived functors from abelian categories. This paper rigorously delves into the precise nature of this exact structure, proposing and demonstrating methodologies to explicitly unveil the abelian foundations that frequently give rise to these triangulated categories. We systematically investigate the relationship between exact categories, Frobenius categories, and their associated stable categories, demonstrating how the latter, often being triangulated, retain crucial information about the former. Special attention is given to the construction of exact sequences and distinguished triangles, exploring how the homological properties encoded within an abelian category manifest in its derived or stable category. Through rigorous theoretical analysis and constructive proofs, we provide a clearer understanding of how the intricate geometry and algebra of triangulated categories are ultimately rooted in more conventional abelian structures, offering new perspectives for their classification and manipulation. Our work makes explicit the processes by which abelian exactness translates into triangulated axioms and shows how abelian ``hearts'' can be extracted from complex triangulated structures. |
| title | The Exact Structure of Triangulated Categories: Unveiling Abelian Foundations |
| url | https://doi.org/10.5281/zenodo.17809014 |