---
tag: X2AHW5VJZR
kind: concept
name: Stable ∞-category
msc: [18N60, 18G80, 55P42]
specializes: [YFY1FWA1MY]
sources:
  - url: https://en.wikipedia.org/wiki/Stable_%E2%88%9E-category
same_as:
  wikidata: Q17073594
  wikipedia: Stable ∞-category
---

## Summary

A stable ∞-category is an [[(∞,1)-category|YFY1FWA1MY]] with a [[zero object|AQET9YHH6F]] and with fibers and cofibers of all morphisms.
In it, fiber sequences and cofiber sequences coincide.
Its homotopy category is a [[triangulated category|T2V7GJJ2EN]] in a natural way.
Stable ∞-categories give a well-behaved replacement for triangulated categories: cones are functorial, and all finite limits and finite colimits exist.
Main examples are the ∞-category of [[spectra|1GHF4EKBXS]] and the derived ∞-category of a ring.

## Explanations

### L1 human

In a stable ∞-category, every map has a kernel-like object, the fiber, and a cokernel-like object, the cofiber.
Stability says that these two constructions fit together perfectly: a sequence is exact from the left exactly when it is exact from the right.
So the loop and suspension operations are inverse to each other, as in the world of chain complexes.
For chain complexes, the cofiber of a map is its mapping cone, and the fiber is the cone shifted down by one.
Pointed topological spaces are not stable: the loop space of the suspension of a space is in general not the space itself.