From Matrices to Morphisms II: Examples of Computational Categories in MATLAB and Octave

Abstract

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.