Polygon of recollements and $N$-complexes

Abstract: We study a structure of subcategories which are called a polygon of recollements in a triangulated category. First, we study a $2n$-gon of recollements in an $(m/n)$-Calabi-Yau triangulated category. Second, we show the homotopy category $\mathsf{K}(\mathsf{Mor}{N-1}(\mathcal{B}))$ of complexes of an additive category $\mathsf{Mor}{N-1}(\mathcal{B})$ of $N-1$ sequences of split monomorphisms of an additive category $\mathcal{B}$ has a $2N$-gon of recollments. Third, we show the homotopy category $\mathsf{K}{N}(\mathcal{B})$ of $N$-complexes of $\mathcal{B}$ has also a $2N$-gon of recollments. Finally, we show there is a triangle equivalence between $\mathsf{K}(\mathsf{Mor}{N-1}(\mathcal{B}))$ and $\mathsf{K}_{N}(\mathcal{B})$.