DefinitionDefinition: 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.

Rotation of a triangle

Definition

The left rotation of $X \overset{f}{\to} Y \overset{g}{\to} Z \overset{h}{\to} X [1]$ is $Y \overset{g}{\to} Z \overset{h}{\to} X [1] \overset{- f [1]}{\to} Y [1] .$ Its inverse (right) rotation is $Z [- 1] \overset{- h [- 1]}{\to} X \overset{f}{\to} Y \overset{g}{\to} Z,$ where the displayed source and target use the specified coherence isomorphisms for $[- 1]$ and $[1]$ . This fixes the sign convention used below.

Depends on

Triangle in a category with translation

Used by

Triangulated-category axiom TR2 Definition
Triangulated-category axiom TR4 (octahedral) Definition
The rotation of a distinguished triangle has no sign False statement
Cone triangles satisfy TR2 with the declared rotation sign Lemma
Distinguished triangles are closed under shifts and both rotations Proposition
The triangulated rotation-sign convention Remark

Dependency tree · two levels

2 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)