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Stable Infinity-Categories

This entry is about the text

• Jacob Lurie, Derived Algebraic Geometry I: Stable $\mathrm{\infty }$-Categories, (arXiv, pdf)

For a definition of stable $\mathrm{\infty }$-categories see stable ∞-category.

Summary

Based on the theory of (∞,1)-categories as developed in his book Higher Topos Theory, Lurie studies here $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)$-category is shown to be a triangulated category: the comparatively complicated axioms of triangulated categories follow from the simple $(\mathrm{\infty },1)$-categorical axioms. Large chunks of homological algebra is then re-examined from the more natural point of view of stable $(\mathrm{\infty },1)$-categories.

Content

1 Introduction

2 Stable $\mathrm{\infty }$-Categories

• stable (∞,1)-category

  • zero object

  • kernel

  • cokernel

  • homotopy fiber

  • fibration sequence

  • loop space object

  • homotopy limit

3 The Homotopy Category of a Stable $\mathrm{\infty }$-Category

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

• triangulated category

4 Properties of Stable $\mathrm{\infty }$-Categories

5 Exact Functors

• exact functor

6 t-Structures and Localization

7 Boundedness and Completeness

8 Stabilization

• stabilization

• spectrum object

9 The $\mathrm{\infty }$-Category of Spectra

• spectrum

• stable (∞,1)-category of spectra

10 Excisive Functors

11 Filtered Objects and Spectral Sequences

12 The $\mathrm{\infty }$-Categorical Dold-Kan Correspondence

• Dold-Kan correspondence

13 Homological Algebra

• homological algebra

• derived category

• derived functor on a derived category

14 The Universal Property of $D^{-}(A)$

15 Presentable Stable $\mathrm{\infty }$-Categories

• accessible (∞,1)-category

• presentable (∞,1)-category

16 Accessible t-Structures

category: reference