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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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.

Depends on

Used by

Comparison of the identity is homotopic to the identity Proposition
Left derived maps preserve composition Proposition

Dependency tree · two levels

9 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)