Triangulated categories

前置知识

41 项必需的前置知识（一直追溯到底）

你将学会

• State Verdier's axioms and derive basic consequences, such as long exact Hom sequences.
• Show that K(A) and D(A) are triangulated.
• Define t-structures and identify A with the heart of the standard t-structure on D(A).

定义

• A triangulated category is an additive category with an autoequivalence [1] and a class of distinguished triangles X → Y → Z → X[1] satisfying Verdier's axioms, including the octahedral axiom.
• A t-structure is a pair of full subcategories (D^{≤0}, D^{≥0}) satisfying truncation axioms; its heart D^{≤0} ∩ D^{≥0} is an abelian category.

定理

• Hom(W, −) sends distinguished triangles to long exact sequences of abelian groups.
• The heart of every t-structure is abelian (Beilinson, Bernstein and Deligne).
• Cones are unique only up to non-unique isomorphism, a defect that motivates enhancements such as dg-categories and stable ∞-categories.

常见误解

• Thinking the cone construction is functorial in a triangulated category; in general it is not.
• Believing every triangulated category is the homotopy category of a stable ∞-category; Muro, Schwede and Strickland found triangulated categories without such models.

通向

延伸阅读

• Amnon Neeman, Triangulated Categories (2001)
• Sergei I. Gelfand and Yuri I. Manin, Methods of Homological Algebra