从矩阵到态射 II：MATLAB 与 Octave 中的计算范畴示例
From Matrices to Morphisms II: Examples of Computational Categories in MATLAB and Octave
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中文总结 AI 辅助
本文提出计算范畴概念，以 MATLAB、Octave 数据结构表示多种数学结构，构建统一框架并实现算法化，为范畴论与矩阵类科学计算提供双向视角。
中文摘要 AI 辅助
本文受科学计算中出现的诸多数学结构已具备具体算法表示这一观察结果的启发，发展了计算范畴的概念。有限集、有限维向量空间、关系和图均可通过 MATLAB 与 Octave 的基础数据结构（如索引向量、矩阵、逻辑数组和稀疏矩阵）表示。核心思想是，范畴常可通过由自然数索引的典范代表元范畴描述，该视角自然引出代表元范畴，其对象为维度，态射为有限计算数据结构。所得框架统一了多个示例，包括矩阵范畴、索引范畴、列空间范畴、有限集的行基模型，以及通过混合态射结合有限子集与有限维向量空间的范畴。本文还引入了带特殊非有限代表元的计算范畴，在保留具体坐标描述的同时扩展了有限情形。全文将范畴构造解释为算法，并利用 MATLAB 与 Octave 的标准操作实现。该工作为范畴论提供了计算视角，反之亦然，为面向矩阵的科学计算提供了范畴论层面的理解。
英文摘要
This article develops the notion of a computational category, motivated by the observation that many mathematical structures occurring in scientific computing already admit concrete algorithmic representations. Finite sets, finite-dimensional vector spaces, relations, and graphs may all be represented through elementary MATLAB and Octave data structures such as indexing vectors, matrices, logical arrays, and sparse matrices. The central idea is that a category may often be described through a category of canonical representatives indexed by the natural numbers. This viewpoint leads naturally to representative categories whose objects are dimensions and whose morphisms are finite computational data structures. The resulting framework unifies several examples, including matrix categories, index categories, categories of column spaces, row-based models of finite sets, and categories combining finite subsets with finite-dimensional vector spaces through mixed morphisms. The paper also introduces computational categories with distinguished non-finite representatives, extending the finite setting while retaining concrete coordinate descriptions. Throughout, categorical constructions are interpreted algorithmically and implemented using standard MATLAB and Octave operations. The work contributes to a computational perspective on category theory and, conversely, to a categorical understanding of matrix-oriented scientific computing.
