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

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.

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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 $1_{P_{∙}} \circ 1_{P_{∙}} = 1_{P_{∙}}$ .

Depends on

Used by

Left derived maps preserve identities Proposition

Dependency tree · two levels

8 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 (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)