Caminos de investigaciónCategory TheoryPlanificado
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
La lección de este tema todavía no se ha escrito. El esquema siguiente forma parte del mapa del plan de estudios y no ha sido revisado.
Se basa en
Requisitos previos obligatorios, hasta el fondo: 63
Aprenderás a
- 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.
Definición
- 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.
Teorema
- 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.
Errores frecuentes
- Thinking stable ∞-categories are triangulated categories under a new name; they carry strictly more information.
Lecturas complementarias
- Jacob Lurie, Higher Algebra — Book-length manuscript, distributed online by the author.
- Jacob Lurie, Higher Topos Theory (2009)