PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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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.

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Comparison maps respect composition up to homotopy

Statement

Assume the Axiom of Dependent Choice.

Given composable morphisms $A \overset{u}{\to} B \overset{v}{\to} C$ and chosen projective resolutions of the three objects, any comparison map lifting $v u$ is chain-homotopic to the composite of a comparison map lifting $u$ with a comparison map lifting $v$ .

Facts & Assumptions

Given: Projective resolutions of $A$ , $B$ , and $C$ , together with comparison maps lifting $u$ and $v$ .

[L1]

Comparison maps exist for the composite morphism (Projective comparison maps exist).

[L2]

Two comparison maps lifting the same object morphism are chain-homotopic (Projective comparison maps are unique up to chain homotopy).

Proof

technique · direct

1.1L1givenconstruct

Let $F_{∙}$ lift $u$ and $G_{∙}$ lift $v$ . By [L1], choose a comparison map $H_{∙}$ lifting the composite $v u$ . The composite $G_{∙} F_{∙}$ also lifts $v u$ .

2.1L2step 1.1∎

Since $H_{∙}$ and $G_{∙} F_{∙}$ lift the same morphism $v u$ , [L2] makes them chain-homotopic. This is exactly the claimed compatibility with composition up to homotopy.

Comparison maps respect composition up to homotopy

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources