Triangulated categories arising from n-fold matrix factorizations

Let $\mathcal{A}$ be an additive category and let $T\text{\hspace{0.17em}⁣}:\mathcal{A}\to \mathcal{A}$ be an additive functor equipped with a natural transformation $\omega \text{\hspace{0.17em}⁣}:{\mathrm{I}\mathrm{d}}_{\mathcal{A}}\to T$. We prove that the homotopy category of $n$-fold matrix factorizations of $\omega$, denoted ${\mathrm{H}\mathrm{F}\mathrm{a}\mathrm{c}\mathrm{t}}_n(\mathcal{A},T,\omega )$, admits a natural structure of a right triangulated category. In particular, when $T$ is an automorphism, the homotopy category ${\mathrm{H}\mathrm{F}\mathrm{a}\mathrm{c}\mathrm{t}}_n(\mathcal{A},T,\omega )$ becomes triangulated. Furthermore, if $\mathcal{A}$ is a Frobenius exact category and $T$ is an autoequivalence, we obtain that the category ${\mathrm{F}\mathrm{a}\mathrm{c}\mathrm{t}}_n(\mathcal{A},T,\omega )$ of $n$-fold $(\mathcal{A},T)$-factorizations of $\omega$ is a Frobenius exact category. Consequently, the stable category of the Frobenius exact category ${\mathrm{F}\mathrm{a}\mathrm{c}\mathrm{t}}_n(\mathcal{A},T,\omega )$ is a triangulated category.