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 of the identity is homotopic to the identity

Statement

Assume the Axiom of Dependent Choice.

Any comparison map lifting the identity of a resolved object is homotopic to the identity chain map on that resolution.

Facts & Assumptions

Given: A projective resolution $P_{∙} \to A$ and a comparison map $F_{∙} : P_{∙} \to P_{∙}$ lifting $1_{A}$ .

[L1]

Comparison maps respect composition up to homotopy (Comparison maps respect composition up to homotopy).

[L2]

Comparison maps lifting the identity exist, and the literal identity chain map is one of them (Projective comparison maps exist).

Proof

technique · direct

1.1L2given

By [L2], both $F_{∙}$ and $1_{P_{∙}}$ are comparison maps lifting $1_{A}$ .

2.1L1step 1.1∎

Apply [L1] with both factors equal to the identity morphism on $A$ , taking the chosen lift of the composite to be $F_{∙}$ and the two factor lifts to be the literal identity chain maps. Then $F_{∙}$ is homotopic to $1_{P_{∙}} \circ 1_{P_{∙}} = 1_{P_{∙}}$ .

Comparison of the identity is homotopic to the identity

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources