Research pathwaysCategory TheoryPlanned
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
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
Required prerequisites, all the way down: 41
You will learn to
- 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).
Definition
- 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.
Theorem
- 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.
Common misconceptions
- 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.
Leads to
Further reading
- Amnon Neeman, Triangulated Categories (2001)
- Sergei I. Gelfand and Yuri I. Manin, Methods of Homological Algebra