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In the previous papers we found a direct method to confirm, for any squarematrix, 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 associatedcategories. We determine in this paper the isomorphism classes of reducedcategories that are associated with the matrix 2 of order 1, 2 and 3. Moregenerally, we managed to give upper and lower bounds for the cardinal of theset of all reduced categories associated to the matrix 2 of a given order.

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