Stable Infinity Categories

Abstract

This paper is an expository account of the theory of stable infinity categories. We prove that the homotopy category of a stable infinity category is triangulated, and that the collection of stable infinity categories is closed under a variety of constructions. We also explain how to construct the derived category of an abelian category (with enough projective objects) as the homotopy category of a suitable stable infinity category; moreover, we characterize this stable infinity category by a universal mapping property.

73 Pages; corrected some minor typographical and mathematical errors 2/10/07: added material on filtered objects and spectral sequences. 9/19/07: various minor changes. 5/8/9: Some of the material on constructing stabilizations has been made more explicit

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• 2-Segal objects and the Waldhausen construction
• A Riemann--Hilbert correspondence for infinity local systems
• Algebraic Theories and (Infinity,1)-Categories
• The moduli stack and motivic Hall algebra for the bounded derived category
• Spherical monadic adjunctions of stable infinity categories
• The stable hull of an exact $\infty$-category
• Spherical adjunctions of stable $\infty$-categories and the relative S-construction
• A homotopy theory of coherently commutative monoidal quasi-categories
• A Whirlwind Tour of the World of $(\infty ,1)$-categories
• 2-fold complete Segal operads
• Braid groups and Kleinian singularities
• Non-left-complete derived categories