Derived categories of $N$-complexes

Abstract

We study the homotopy category ${\mathsf{K}}_N(\mathcal{B})$ of $N$-complexes of an additive category $\mathcal{B}$ and the derived category ${\mathsf{D}}_N(\mathcal{A})$ of an abelian category $\mathcal{A}$. First we show that both ${\mathsf{K}}_N(\mathcal{B})$ and ${\mathsf{D}}_N(\mathcal{A})$ have natural structures of triangulated categories. Then we establish a theory of projective (resp., injective) resolutions and derived functors. Finally, under some conditions of an abelian category $\mathcal{A}$, we show that ${\mathsf{D}}_N(\mathcal{A})$ is triangle equivalent to the ordinary derived category $\mathsf{D}({\mathsf{Morph}}_{N-2}(\mathcal{A}))$ where ${\mathsf{Morph}}_{N-2}(\mathcal{A})$ is the category of sequential $N-2$ morphisms of $\mathcal{A}$.

31 pages

References in corpus (1)

• N-complexes as functors, amplitude cohomology and fusion rules

Cited by in corpus (1)

• Categorical matrix factorizations