nLab Stable Infinity-Categories

This entry provides a hyperlinked index for

• Jacob Lurie, Stable $\infty$-Categories, (arXiv:math.CT/0608228), now section 1 in Higher Algebra

on stable homotopy theory in terms of stable ∞-categories.

Based on the theory of (∞,1)-categories as developed in his book Higher Topos Theory, the author studies here $(\infty,1)$-categories of “stable objects”, i.e. of objects that behave like spectra in that for each object $X$ there not only its loop space object $\Omega X$ but also conversely, $X$ is the loop space object of another object $\Sigma X$.

The definition is very simple. The homotopy category of a stable $(\infty,1)$-category is shown to be a triangulated category: the comparatively complicated axioms of triangulated categories follow from the simple $(\infty,1)$-categorical axioms. Large chunks of homological algebra is then re-examined from the more natural point of view of stable $(\infty,1)$-categories.

Introduction

Stable $\infty$-Categories

• stable (∞,1)-category

  • zero object

  • kernel

  • cokernel

  • homotopy fiber

  • fibration sequence

  • loop space object

  • homotopy limit

The Homotopy Category of a Stable $\infty$-Category

• homotopy category of an (∞,1)-category

• triangulated category

Properties of Stable $\infty$-Categories

Exact Functors

• exact functor

t-Structures and Localization

Boundedness and Completeness

Stabilization

• stabilization

• spectrum object

The $\infty$-Category of Spectra

• spectrum

• stable (∞,1)-category of spectra

Excisive Functors

Filtered Objects and Spectral Sequences

The $\infty$-Categorical Dold-Kan Correspondence

• Dold-Kan correspondence

Homological Algebra

• homological algebra

• derived category

• derived functor on a derived category

The Universal Property of $D^-(A)$

Presentable Stable $\infty$-Categories

• accessible (∞,1)-category

• presentable (∞,1)-category

Accessible t-Structures

category: reference