Exceptional model structures and the induced extriangulated categories

Abstract

A Hovey triple $(\mathcal{C},\mathcal{F},\mathcal{W})$ is {\it exceptional}, if $(\mathcal{C}\cap \mathcal{F},\mathcal{C}\cap \mathcal{F}\cap \mathcal{W})$ is not a Frobenius pair. One has a disjoint union $\{\text{Hovey triple}\}=\{\text{Hereditary Hovey triple}\}\dot{\bigcup }\{\text{Exceptional Hovey triple}\}$ \ $\dot{\bigcup }\{\text{non-hereditary and non-exceptional Hovey triple}\}.$ Exceptional Hovey triples appear widely. For a selfinjective Nakayama algebra $A=k{C}_n/J^t$, $A\mbox{-}{\rm mod}$ admits exceptional Hovey triples if and only if $\gcd (n,t)\ge 2$ and $t\ge 3$. Their homotopy categories reveal new phenomena. Nakaoka-Palu's Theorem implies that there is a triangulation on $\frac{\mathcal{C}\cap \mathcal{F}}{\mathcal{C}\cap \mathcal{F}\cap \mathcal{W}}.$ It is proved that a Hovey triple in a weakly idempotent complete extriangulated category $(\mathcal{A},\mathbb{E},\mathfrak{s})$ is exceptional if and only if the induced extriangulated structure $(\frac{\mathcal{C}\cap \mathcal{F}}{\mathcal{C}\cap \mathcal{F}\cap \mathcal{W}},\overline{\mathbb{E}},\overline{\mathfrak{s}})$ is not {\it canonically triangulated}. Between this extriangulated structure and the one arising from the triangulation, the intermediate extriangulated structures are in one-to-one correspondence with Serre subcategories of $\mathrm{fp}((\frac{\mathcal{P}(\mathcal{C}\cap \mathcal{F})}{\mathcal{C}\cap \mathcal{F}\cap \mathcal{W}}{)}^{\mathrm{op}},\mathrm{Ab})$.