Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Triangulated category
Definition
A triangulated category is a category with translation and a class of distinguished triangles satisfying TR1, TR2, TR3, and TR4. Both the translation and the class are specified structure, not properties inferred from the underlying additive category.
Depends on
Used by
- Cohomological functor on a triangulated category Definition
- Exact functor between triangulated categories Definition
- Homological functor on a triangulated category Definition
- Triangulated subcategory Definition
- A map is zero exactly when the corresponding representable map vanishes Proposition
- Distinguished triangles are closed under shifts and both rotations Proposition
- The cone object of a map is unique up to nonunique isomorphism Proposition
- The octahedral axiom gives a triangle relating the cones of f, g, and gf Proposition
- Zero and split triangles are distinguished Proposition
- The triangulated rotation-sign convention Remark
- Long exact Hom sequences of a distinguished triangle Theorem
- Representable Hom functors on a triangulated category are homological or cohomological Theorem
Dependency tree · two levels
10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- The Stacks Project, Definition 13.3.2 (standard reference, not scraped)