Homotopy categories of admissible model structures on extriangulated categories
Shun-Jie Li, Yang Gao, Pu Zhang
TL;DR
This work provides an alternative triangulated-structure proof for the homotopy category of an admissible model structure on a weakly idempotent complete extriangulated category. By constructing standard triangles from suspension sequences and $\mathbb{E}$-triangles, it defines a class $\Delta$ of distinguished triangles and proves that $(\mathrm{Ho}(\mathcal{A}),\Sigma,\Delta)$ is triangulated, offering a direct link to Heller–Happel’s framework. It also clarifies the relationship with NP’s original distinguished triangles by showing $\Delta = -\widetilde{\Delta}$, yielding a triangle-isomorphism between the two triangulations. The results unify model-categorical, cotorsion-pair, and extriangulated techniques, and extend to stable categories of Frobenius extriangulated categories, highlighting broad applicability in homological algebra and representation theory. Overall, the paper provides a streamlined, self-contained alternative route to NP’s theorem and deepens the connections between extriangulated structures and triangulated homotopy theories.
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 $Δ$ of distinguished triangles in the present paper and the class $\widetildeΔ$ of distinguished triangles in \cite{NP} have the relation $Δ= - \widetildeΔ$, and hence the two triangulated structures on the homotopy category are isomorphic.