DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Equivalence of n-fold extensions

Definition

For two $n$ -fold extensions with the same endpoints, begin with chain maps between their displayed exact sequences that are the identity on $N$ and $M$ . Their reflexive, symmetric, and transitive closure is called equivalence of $n$ -fold extensions. For $n > 1$ , this generated relation does not require one isomorphism of all middle terms.

Depends on

An n-fold Yoneda extension

Used by

Equivalent higher extensions need not have one middle-term isomorphism Counterexample
Yoneda splicing is well-defined on equivalence classes Lemma
Higher Yoneda Ext agrees with derived Ext Theorem
The Yoneda product is associative and unital Theorem

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

Charles A. Weibel, An Introduction to Homological Algebra, Chapters 3–4 (standard reference, not scraped)