研究路径Category Theory计划中
Stable ∞-categories
Optional research direction: stable ∞-categories, in which fibre and cofibre sequences coincide, as enhancements of triangulated categories; spectra; and the passage from a stable ∞-category to its triangulated homotopy category.
math.cat.stable-infinity-categories
本主题的课程尚未编写。下面的提纲是课程地图的一部分，尚未经过审校。
前置知识
63 项必需的前置知识（一直追溯到底）
你将学会
- Define stable ∞-categories and give examples, including derived ∞-categories of rings and the ∞-category of spectra.
- Explain why the homotopy category of a stable ∞-category is triangulated.
- Explain how stable ∞-categories resolve the non-functoriality of cones.
定义
- An ∞-category is stable if it has a zero object, every morphism has a fibre and a cofibre, and a triangle is a fibre sequence if and only if it is a cofibre sequence.
- Spectra form the stable ∞-category Sp, the stabilisation of the ∞-category of pointed spaces.
定理
- The homotopy category of a stable ∞-category is canonically triangulated (Lurie, Higher Algebra).
- In a stable ∞-category, cofibres are functorial.
- The derived category of a ring is the homotopy category of a stable ∞-category.
常见误解
- Thinking stable ∞-categories are triangulated categories under a new name; they carry strictly more information.
延伸阅读
- Jacob Lurie, Higher Algebra — Book-length manuscript, distributed online by the author.
- Jacob Lurie, Higher Topos Theory (2009)