词条 | Stable ∞-category |
释义 |
In category theory, a branch of mathematics, a stable ∞-category is an ∞-category such that[1]
The homotopy category of a stable ∞-category is triangulated.[2] A stable ∞-category admits finite limits and colimits.[3] Examples: the derived category of an abelian category and the ∞-category of spectra are both stable. A stabilization of an ∞-category C having finite limits and base point is a functor from the stable ∞-category S to C. It preserves limit. The objects in the image have the structure of infinite loop spaces; whence, the notion is a generalization of the corresponding notion (stabilization (topology)) in classical algebraic topology. By definition, the t-structure of an stable ∞-category is the t-structure of its homotopy category. Let C be a stable ∞-category with a t-structure. Then every filtered object in C gives rise to a spectral sequence , which, under some conditions, converges to [4] By the Dold–Kan correspondence, this generalizes the construction of the spectral sequence associated to a filtered chain complex of abelian groups. Notes1. ^{{harvnb|Lurie|2012|loc=Definition 1.1.1.9.}} 2. ^{{harvnb|Lurie|2012|loc=Theorem 1.1.2.14.}} 3. ^{{harvnb|Lurie|2012|loc=Proposition 1.1.3.4.}} 4. ^{{harvnb|Lurie|2012|loc=Construction 1.2.2.6.}} References
1 : Higher category theory |
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