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 $\mathsf{ K}_{ N}(\mathcal{ B})$.
Categories: math.CT math.RT