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.

Standard cone triangle in the homotopy category

Definition

Let $A$ be an additive category. For a chain map $f : C_{∙} \to D_{∙}$ in $A$ , its standard cone triangle in $K (A)$ is the image of $C_{∙} \overset{f}{\to} D_{∙} \overset{j}{\to} Cone (f)_{∙} \overset{q}{\to} C_{∙} [1]$ under the quotient to the homotopy category. The final map and shift are the ones already specified by the chain-level cone triangle The cone triangle of a chain map.

Depends on

Used by

Distinguished cone triangle in the homotopy category Definition
The cone triangle of multiplication by m Example
The third map in a morphism of triangles is unique False statement
Cone triangles satisfy TR1 Lemma
A quasi-isomorphism has an acyclic cone in the triangulated language Proposition
An additive functor on abelian categories induces an exact functor on homotopy categories Proposition

Dependency tree · two levels

6 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 13.9.1 (standard reference, not scraped)