Caminos de investigaciónCategory TheoryPlanificado
Triangulated categories
Optional research direction: additive categories with a shift functor and distinguished triangles satisfying Verdier's axioms, including the octahedral axiom; examples from derived and stable homotopy categories; t-structures and their hearts.
math.cat.triangulated-categories
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Se basa en
Requisitos previos obligatorios, hasta el fondo: 41
Aprenderás a
- 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).
Definición
- 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.
Teorema
- 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.
Errores frecuentes
- 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.
Conduce a
Lecturas complementarias
- Amnon Neeman, Triangulated Categories (2001)
- Sergei I. Gelfand and Yuri I. Manin, Methods of Homological Algebra