Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07
How 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.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- 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.
Exact functor between triangulated categories
Definition
An exact functor is an additive functor with a specified natural isomorphism such that every distinguished has distinguished image .
Depends on
Used by
- Triangulated subcategory Definition
- Bounded two-sided projective bimodule complexes act on Cₘ Lemma
- Shift signs and exact-functor maps on triangulated K0 Lemma
- A translation-compatible natural isomorphism of exact functors respects triangles Proposition
- An additive functor on abelian categories induces an exact functor on homotopy categories Proposition
- A bounded two-sided projective bimodule complex defines exact derived tensor functors Theorem
- Graded derived tensor equivalences induce Laurent-linear K0 and G0 maps Theorem
- The derived category inherits a triangulated structure Theorem
Dependency tree · two levels
15 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 13.3.3 (standard reference, not scraped)