Stable ∞-category

In category theory, a branch of mathematics, a stable ∞-category is an ∞-category such that

The homotopy category of a stable ∞-category is triangulated. A stable ∞-category admits finite limits and colimits.

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 ${\displaystyle X(i),i\in \mathbb {Z} }$ in C gives rise to a spectral sequence ${\displaystyle E_{r}^{p,q}}$, which, under some conditions, converges to ${\displaystyle \pi _{p+q}\operatorname {colim} X(i).}$ By the Dold–Kan correspondence, this generalizes the construction of the spectral sequence associated to a filtered chain complex of abelian groups.

...
Wikipedia