Alphabeta Math
TheoremStatement: 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.

Injective resolutions of the same object are homotopy equivalent under that object

Statement

Any two injective resolutions of the same object are homotopy equivalent under that object.

Facts & Assumptions

Given: Two injective resolutions I∙ and J∙ of the same object A.

[L1]

Injective comparison maps exist (Injective comparison maps exist).

[L2]

Injective comparison maps are unique up to cochain homotopy (Injective comparison maps are unique up to cochain homotopy).

Proof

technique · direct
1.1

Apply [L1] to the identity on A in both directions. This yields maps I∙→J∙ and J∙→I∙ extending 1A.

L1construct
2.1

Their composites and the identity cochain maps all extend 1A, so [L2] makes the composites homotopic to the identities. Hence the two injective resolutions are homotopy equivalent under A, including when A=0.

L2step 1.1∎

Depends on

Used by

Dependency tree · two levels

6 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