शोध पथCategory Theoryनियोजित
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
इस विषय का पाठ अभी लिखा नहीं गया है। नीचे दी गई रूपरेखा पाठ्यक्रम-मानचित्र का हिस्सा है और इसकी समीक्षा नहीं हुई है।
इस पर आधारित
63 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)
आप सीखेंगे
- 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.
परिभाषा
- 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.
प्रमेय
- 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.
आम भ्रांतियाँ
- Thinking stable ∞-categories are triangulated categories under a new name; they carry strictly more information.
आगे पढ़ने के लिए
- Jacob Lurie, Higher Algebra — Book-length manuscript, distributed online by the author.
- Jacob Lurie, Higher Topos Theory (2009)