Triadic categories without localization

Triadic categories arise as homotopy categories M(Λ-CM) of 2-termed complexes over a category Λ-CM of representations of a finite dimensional algebra, a classical order, or any higher-dimensional Cohen–Macaulay order Λ. They are close to triangulated categories, with the main difference that triangles are replaced by 4-termed complexes (= triads) which are functorial. In addition, the triadic category M(Λ-CM) is exact, i.e. it has a distinguished class of short exact sequences with the usual properties. We give a new characterization of triadic categories which makes no use of localization. As a consequence, the triadic structure of M(Λ-CM) can be derived from the exact structure of M(M(Λ-CM)).