Stable ∞-category

In , a branch of mathematics, a stable ∞-category is an such that (i) It has a zero object. (ii) Every morphism in it admits a and cofiber. (iii) A triangle in it is a fiber sequence if and only if it is a cofiber sequence. The of a stable ∞-category is . A stable ∞-category admits finite s and colimits. Examples: the of an and the ∞-category of spectra are both stable. A stabilization of an 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 a 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 By the Dold–Kan correspondence, this generalizes the construction of the spectral sequence associated to a filtered chain complex of abelian groups.