Homotopy categories of admissible model structures on extriangulated categories

TL;DR

This paper offers an alternative proof that the homotopy category of an admissible model structure on an extriangulated category is triangulated, connecting different constructions of distinguished triangles.

Contribution

It provides a new proof of Nakaoka and Palu's theorem using classical constructions, showing the equivalence of two triangulated structures.

Findings

The homotopy category is triangulated.

Different classes of distinguished triangles are shown to be related.

The two triangulated structures are isomorphic.

Abstract

The extriangulated category is a simultaneous generalization of exact categories and triangulated categories. H. Nakaoka and Y. Palu have proved that the homotopy category of an admissible model structure on a weakly idempotent complete extriangulated category is a triangulated category. Using the classic construction of distinguished triangles given by A. Heller and D. Happel, this paper provides an alternative proof of Nakaoka - Palu Theorem. In fact, the class $\Delta$ of distinguished triangles in the present paper and the class $\widetilde{\Delta}$ of distinguished triangles in \cite{NP} have the relation $\Delta = - \widetilde{\Delta}$, and hence the two triangulated structures on the homotopy category are isomorphic.