DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Minimal A∞-model

Definition

Let $A$ be an uncurved A∞-algebra (Uncurved cohomological A∞-algebra in the unsuspended convention). A minimal A∞-model for $A$ is an uncurved A∞-algebra $H$ with $m_{1}^{H} = 0$ , together with an A∞ quasi-isomorphism $f : H \to A$ . Here an A∞ quasi-isomorphism is as defined in A∞-morphism and quasi-isomorphism. In an explicit model whose underlying graded space is identified with $H^{*} (A, m_{1}^{A})$ , the binary product is ordinarily checked against the product induced by $m_{2}^{A}$ ; this definition does not assert an existence or transfer theorem.

This is a definition of the data required for a minimal model. It does not assert that every A∞-algebra has one or that a transfer theorem has been proved.

Depends on

Used by

A minimal A∞-model whose ternary operation realizes a triple Massey product Example

Dependency tree · two levels

4 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

Bernhard Keller, Introduction to A-infinity algebras and modules, §3.3 (standard reference, not scraped)
Jean-Louis Loday and Bruno Vallette, Algebraic Operads, §10.3 (standard reference, not scraped)