Classification des cat\'egories associ\'ees \`a la matrice des coefficients 2 d'ordre donn\'ee

TL;DR

This paper classifies and bounds the number of reduced categories associated with the all-2 coefficient matrix for matrices of order 1, 2, and 3, extending previous methods for matrix-category association.

Contribution

It determines isomorphism classes for small matrices and provides bounds for the set size of all associated reduced categories for any order.

Findings

Classified reduced categories for matrices of order 1, 2, and 3.

Established bounds for the number of associated reduced categories.

Extended previous methods to general matrix orders.

Abstract

In the previous papers we found a direct method to confirm, for any square matrix, if it is associated to any categories or not. According to this method, the matrix 2 (all coefficients are 2) of a given order, admits associated categories. We determine in this paper the isomorphism classes of reduced categories that are associated with the matrix 2 of order 1, 2 and 3. More generally, we managed to give upper and lower bounds for the cardinal of the set of all reduced categories associated to the matrix 2 of a given order.