AI 中文总结
本文推广Cole定理，在弱幂等完备外三角范畴中从两个相容的适当模型结构构造混合容许模型结构，刻画其余纤维对象，并应用于正合与三角范畴以推广近期结果。
AI 中文摘要
设 $(\mathscr{C},\mathbb{E},\mathfrak{s})$ 是一个弱幂等完备的外三角范畴。我们将 Cole 定理推广，从相对于 $\mathscr{C}$ 的适当类 $\xi_1\subseteq\xi_2$ 的两个相容的适当模型结构出发，构造一个混合容许模型结构 $\mathcal{M}_m$。然后我们明确刻画了 $\mathcal{M}_m$ 的余纤维对象。最后，我们将这些结果应用于正合范畴和三角范畴，恢复并推广了关于混合模型结构的近期工作。
英文摘要
Let $(\mathscr{C}, \mathbb{E}, \mathfrak{s})$ be a weakly idempotent complete extriangulated category, and let $ξ_1\subseteqξ_2$ be proper classes of $\mathbb{E}$-triangles. We prove an analogue of Cole's mixing theorem in this setting: two compatible admissible model structures, relative to $ξ_1$ and $ξ_2$ respectively, give rise to a mixed admissible model structure $\mathcal{M}_m$ relative to $ξ_1$, which has the trivial objects of the second model structure and the fibrant (or cofibrant) objects of the first. We also describe the cofibrant objects of $\mathcal{M}_m$ explicitly in terms of the two original model structures. For exact categories this recovers the mixing theorem for exact model structures, and for triangulated categories it applies to proper classes of triangles; we give a non-degenerate example on the homotopy category of a quasi-Frobenius ring.
Comments18 pages. Revised exposition and removed an AI-generated example
