DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-08-31

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.

The mapping cone of a chain map

Definition

Let $f : C_{∙} \to D_{∙}$ be a chain map in an additive category. The mapping cone of $f$ is the chain complex $Cone (f)_{∙}$ whose degree- $n$ term is $Cone (f)_{n} : = D_{n} \oplus C [1]_{n} = D_{n} \oplus C_{n - 1},$ and whose differential is $d_{n}^{Cone (f)} (y, x) : = (d_{n}^{D} (y) + f_{n - 1} (x), - d_{n - 1}^{C} (x)) .$

Thus $Cone (f)$ is the direct sum of $D$ and the shift $C [1]$ as a graded object, with the off-diagonal term given by $f$ .

The mapping cone of a chain map

Definition

Depends on

Dependency tree · 2 levels

Sources