View source for Stable ∞-category

In [[Category theory|category theory]], a branch of mathematics, a '''stable ∞-category''' is an ∞-category such that<ref>{{harvnb|Lurie|loc=Definition 1.1.1.9.}}</ref>
*(i) It has a zero object.
*(ii) Every [[Morphism|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|homotopy category]] of a stable ∞-category is [[Triangulated category|triangulated]].<ref>{{harvnb|Lurie|loc=Theorem 1.1.2.14.}}</ref> A stable ∞-category admits finite [[Limit (category theory)|limit]]s and colimits.<ref>{{harvnb|Lurie|loc=Proposition 1.1.3.4.}}</ref>

Examples: the [[Derived category|derived category]] of an [[Abelian category|abelian category]] and the ∞-category of [[Spectrum (topology)|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|algebraic topology]].

By definition, the [[T-structure|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 <math>X(i), i \in \mathbb{Z}</math> in ''C'' gives rise to a [[Spectral sequence|spectral sequence]] <math>E^{p, q}_r</math>, which, under some conditions, converges to <math>\pi_{p+q} \operatorname{colim} X(i).</math><ref>{{harvnb|Lurie|loc=Construction 1.2.2.6.}}</ref> By the [[Dold–Kan correspondence]], this generalizes the construction of the [[Spectral sequence|spectral sequence]] associated to a filtered [[Chain complex|chain complex]] of [[Abelian group|abelian group]]s.

== Notes ==
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== References ==
*{{Lurie-HA}}

{{Category theory}}

[[Category:Higher category theory]]


{{Sourceattribution|Stable ∞-category}}