Triangulated categories arising from n -fold matrix factorizations

Let [Formula: see text] be an additive category and let [Formula: see text] be an additive functor equipped with a natural transformation [Formula: see text]. We prove that the homotopy category of [Formula: see text]-fold matrix factorizations of [Formula: see text], denoted [Formula: see text], admits a natural structure of a right triangulated category. In particular, when [Formula: see text] is an automorphism, the homotopy category [Formula: see text] becomes triangulated. Furthermore, if [Formula: see text] is a Frobenius exact category and [Formula: see text] is an exact autoequivalence, we obtain that the category [Formula: see text] of [Formula: see text]-fold [Formula: see text]-factorizations of [Formula: see text] is a Frobenius exact category. Consequently, the stable category of the Frobenius exact category [Formula: see text] is a triangulated category.