Polygon of recollements and $N$-complexes

TL;DR

This paper explores polygonal structures of recollements in triangulated categories, demonstrating their existence in categories of N-complexes and establishing an equivalence between certain homotopy categories.

Contribution

It introduces the concept of polygons of recollements in triangulated categories and proves their presence in categories of N-complexes, along with an equivalence result.

Findings

Existence of 2n-gons of recollements in (m/n)-Calabi-Yau categories

Homotopy category of complexes of N-1 sequences has a 2N-gon of recollements

Homotopy category of N-complexes also has a 2N-gon of recollements

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 $\mathsf{K}_{N}(\mathcal{B})$.