正交模型结构
Orthogonal Model Structures
- School of Mathematical Sciences Shanghai Jiao Tong University（上海交通大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究正交模型结构，证明其同伦范畴等价于余纤维-纤维对象全子范畴，并建立TTF、双反射及挠模型结构与相应三元组、对及孪生挠对的构造性一一对应，且计算了各类模型结构的同伦范畴。
AI中文摘要：
本文研究正交模型结构，即提升公理中的提升具有唯一性的模型结构。余纤维（纤维）对象在不依赖初始（终）对象的情况下定义。对于具有足够余纤维对象和纤维对象的范畴上的正交模型结构，证明了同伦范畴等价于余纤维-纤维对象的全子范畴。引入了TTF模型结构、双反射模型结构和挠模型结构，它们都是正交的。在阿贝尔范畴中，建立了TTF模型结构与TTF三元组之间的一一对应；在任意范畴中，建立了双反射模型结构与双反射对之间的一一对应；在阿贝尔范畴中，建立了挠模型结构与孪生挠对之间的一一对应，且这些对应均以构造性方式给出。还构造了偏序集范畴、$G$-集合范畴和拓扑群范畴上的挠模型结构。计算了所有这些模型结构的同伦范畴。特别地，TTF模型结构的同伦范畴是阿贝尔范畴。
英文摘要：
This paper studies orthogonal model structures, i.e., model structures such that a lifting in the Lifting axiom is unique. Cofibrant (fibrant) objects are defined without initial (terminal) objects. For an orthogonal model structure on a category with enough cofibrant objects and fibrant objects, it is proved that the homotopy category is equivalent to the full subcategory of cofibrant-fibrant objects. TTF model structures, bi-reflective model structures, and torsion model structures, are introduced. They are all orthogonal. One to one correspondences between TTF model structures and TTF triples in an abelian category, bi-reflective model structures and bi-reflective pairs in any category, and torsion model structures and twin torsion pairs in an abelian category, are established in a constructive way. Torsion model structures on a poset, on the category of $G$-sets, and on the category of topological groups, are also constructed. The homotopy categories of all these model structures are computed. In particular, the homotopy category of a TTF model structure is an abelian category.
