Homotopical Foundations for Linear Algebra Article Swipe

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SÉRGIO DE ANDRADE, PAULO · YOU? · · 2025 · Open Access · · DOI: https://doi.org/10.5281/zenodo.17689498

This paper explores the establishment of homotopical foundations for linear algebra, moving beyond the traditional set-theoretic and categorical approaches to incorporate concepts from homotopy theory. We propose a framework where vector spaces, linear transformations, and their fundamental properties are reinterpreted within a higher categorical context, emphasizing homotopy equivalences over strict isomorphisms. The study begins by reviewing the historical development of linear algebra and the emergence of homological algebra as a tool for studying algebraic structures with greater depth. We then introduce core concepts from category theory and homotopy theory, such as model categories, chain complexes, and derived categories, demonstrating their relevance in providing a more robust and flexible foundation. Our methodology involves defining homotopical analogues of classical linear algebraic constructions, such as tensor products, direct sums, and dual spaces, and investigating their properties up to homotopy. This perspective allows for a richer understanding of algebraic invariants and resolutions, providing insights into the stability and deformation properties of linear algebraic systems. The results section presents explicit examples of how theorems like the rank-nullity theorem or the structure theorem for finitely generated modules can be re-contextualized within a homotopical setting, leading to potentially new generalizations and computational approaches. The discussion highlights the implications of this framework for algebraic topology, algebraic geometry, and theoretical physics, where linear algebra plays a crucial role but often benefits from a more flexible, homotopy-invariant formulation. We conclude by outlining future research directions, particularly in developing a full-fledged model category structure for vector spaces and exploring applications in areas such as persistent homology and quantum information theory.

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article

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https://doi.org/10.5281/zenodo.17689498

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https://openalex.org/W7106550376

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https://doi.org/10.5281/zenodo.17689498

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Homotopical Foundations for Linear Algebra

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article

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2025

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2025-11-23

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SÉRGIO DE ANDRADE, PAULO

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https://doi.org/10.5281/zenodo.17689498

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Mathematics, Algebra over a field, Homotopy, Homological algebra, Categorical variable, Linear algebra, Algebraic structure, Algebraic number, Homology (biology), Vector space, Tensor algebra, Pure mathematics, Category theory, Algebraic geometry, Dimension of an algebraic variety, Algebraic topology, Vector bundle, Dual (grammatical number), Universal algebra, Homotopy category, Algebraic cycle, Algebra representation, Perspective (graphical), Tensor product, Linear map, Commutative algebra, Relevance (law)

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