Questions: Triangulated Categories

Question 1

What is the primary structural role of distinguished triangles in a triangulated category?

• A. They provide a multiplication structure that makes the category into a ring
• B. They generalize short exact sequences, generating long exact sequences when a cohomological functor is applied
• C. They replace morphisms with higher-dimensional cells, extending the category to an ∞-category
• D. They classify all objects up to isomorphism using three canonical invariants

Answer: (B)

In an abelian category, short exact sequences 0 → A → B → C → 0 generate long exact sequences in cohomology. Triangulated categories extend this to settings — like derived categories — where 'sub-object' and 'quotient' may not exist. The distinguished triangle A → B → C → ΣA plays the role of a short exact sequence: applying a cohomological functor H produces the long exact sequence ⋯ → H(A) → H(B) → H(C) → H(ΣA) → ⋯. This computational payoff is the main reason for working in triangulated categories.

Question 2

In the derived category D(𝒜) of an abelian category, a short exact sequence 0 → A → B → C → 0 gives rise to which structure?

• A. A direct sum decomposition B ≅ A ⊕ C in D(𝒜)
• B. A distinguished triangle A → B → C → ΣA in D(𝒜)
• C. A new abelian category whose objects are the exact sequences themselves
• D. A chain homotopy equivalence between A⊕C and B

Answer: (B)

Short exact sequences in the abelian category 𝒜 pass to distinguished triangles in D(𝒜). This is the precise way in which the derived category extends abelian homological algebra: exact sequences become triangles, and the long exact sequence in cohomology is recovered from the triangle. If the sequence split (B ≅ A⊕C), the triangle would be split distinguished — but this is a special case, not the general one.

Question 3

Rotating a distinguished triangle A → B → C → ΣA generally produces a triangle that is no longer distinguished.

• T. True
• F. False

Answer: False

Rotation is an axiom of triangulated categories. If A → B → C → ΣA is distinguished, then B → C → ΣA → ΣB is also distinguished (and so on by further rotation). Rotation shifts perspective around the triangle — what was B becomes A, what was C becomes B, what was ΣA becomes C — reflecting the symmetry between the roles of sub-object, object, and quotient in the underlying homological structure.

Question 4

The octahedral axiom ensures that given composable morphisms f: A → B and g: B → C, the cofibers of f, g, and g∘f fit into a coherent distinguished triangle.

• T. True
• F. False

Answer: True

The octahedral axiom states that if you know how B is built from A (via f), how C is built from B (via g), and how C is built directly from A (via g∘f), the three cofibers — Cone(f), Cone(g), Cone(g∘f) — are themselves related by a distinguished triangle. This coherence condition ensures the triangulated structure is compatible with composition. In D(𝒜), the axiom follows from the Snake Lemma in 𝒜, making it geometrically natural even when it appears formally mysterious.

Question 5

Explain how distinguished triangles in a triangulated category serve the same purpose as short exact sequences in an abelian category, and what the cohomological functor provides in this setting.

Model answer: In an abelian category, short exact sequences 0→A→B→C→0 encode extension data and generate long exact sequences in any cohomological functor H: ⋯→H^n(A)→H^n(B)→H^n(C)→H^{n+1}(A)→⋯. Triangulated categories generalize this: a distinguished triangle A→B→C→ΣA plays the role of the short exact sequence (C is the 'cofiber' or categorical cokernel of A→B; ΣA is the suspension shifting degree by 1). Applying a cohomological functor H to the triangle produces the same long exact sequence pattern, giving the same computational power even in contexts — like stable homotopy theory or derived categories of sheaves — where sub-objects and quotients do not exist as abelian category objects.

The distinguished triangle is the minimal structure needed to produce long exact sequences. The octahedral axiom then ensures these long exact sequences are compatible when you compose morphisms — the same coherence that the Snake Lemma provides in abelian categories. This is why triangulated categories are the natural setting for derived functors, Tor, Ext, and sheaf cohomology in algebraic geometry.