Mixing Extriangulated Model Structures

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be a weakly idempotent complete extriangulated category. We generalize Cole’s Theorem to construct a mixed admissible model structure ${\mathcal{M}}_m$ from two compatible admissible model structures relative to proper classes ${\xi }_1\subseteq {\xi }_2$ of $\mathcal{C}$. We then explicitly characterize the cofibrant objects of ${\mathcal{M}}_m$. Finally, we apply these results to exact and triangulated categories, recovering and extending recent work on mixed model structures.