Triangulated categories arising from n-fold matrix factorizations

TL;DR

This paper studies the structure of categories arising from n-fold matrix factorizations, showing they form right triangulated or Frobenius categories, and become triangulated under certain conditions, extending the theory of matrix factorizations.

Contribution

It introduces a new framework for n-fold matrix factorizations, establishing their homotopy categories as right triangulated or Frobenius categories, and identifies conditions for triangulated structures.

Findings

Homotopy category of n-fold matrix factorizations is right triangulated.

When T is an automorphism, the category becomes triangulated.

Frobenius structure arises when a0A is Frobenius and T is an autoequivalence.

Abstract

Let $\mathcal{A}$ be an additive category and let $T\colon \mathcal{A}\rightarrow \mathcal{A}$ be an additive functor equipped with a natural transformation $\omega\colon \mathrm{Id}_{\mathcal{A}}\rightarrow T$. We prove that the homotopy category of $n$-fold matrix factorizations of $\omega$, denoted ${\rm HFact}_{n}(\mathcal{A},T,\omega)$, admits a natural structure of a right triangulated category. In particular, when $T$ is an automorphism, the homotopy category ${\rm HFact}_{n}(\mathcal{A},T,\omega)$ becomes triangulated. Furthermore, if $\mathcal{A}$ is a Frobenius exact category and $T$ is an autoequivalence, we obtain that the category ${\rm Fact}_{n}(\mathcal{A},T,\omega)$ of $n$-fold $(\mathcal{A},T)$-factorizations of $\omega$ is a Frobenius exact category. Consequently, the stable category of the Frobenius exact category ${\rm Fact}_{n}(\mathcal{A},T,\omega)$ is a triangulated category.