Stable ∞-category

In category theory, a branch of mathematics, a stable ∞-category is an ∞-category such that^[1]

• (i) It has a zero object.
• (ii) Every morphism in it admits a fiber and cofiber.
• (iii) A triangle in it is a fiber sequence if and only if it is a cofiber sequence.

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 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 ${\displaystyle X(i),i\in \mathbb {Z} }$ in C gives rise to a spectral sequence ${\displaystyle E_{r}^{p,q}}$, which, under some conditions, converges to ${\displaystyle \pi _{p+q}\operatorname {colim} X(i).}$^[4] By the Dold–Kan correspondence, this generalizes the construction of the spectral sequence associated to a filtered chain complex of abelian groups.

Notes

1. Lurie, Definition 1.1.1.9.
2. Lurie, Theorem 1.1.2.14.
3. Lurie, Proposition 1.1.3.4.
4. Lurie, Construction 1.2.2.6.

References

• Lurie, J. "Higher Algebra" (PDF). last updated August 2017