Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 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.

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

A projective or injective resolution is unique up to nonunique homotopy equivalence

Statement

A projective resolution or an injective resolution of a fixed object is unique up to homotopy equivalence, but the chosen comparison maps need not be unique.

Facts & Assumptions

Given: A fixed object A.

[L1]

Projective resolutions of A are homotopy equivalent over A (Projective resolutions of the same object are homotopy equivalent over that object).

[L2]

Injective resolutions of A are homotopy equivalent under A (Injective resolutions of the same object are homotopy equivalent under that object).

Proof

technique · direct
1.1

The projective statement is exactly [L1], and the injective statement is exactly [L2].

L1L2
2.1

Thus either kind of resolution is unique only up to homotopy equivalence. The preceding comparison theorems show that one may choose many actual lifts inside that homotopy class, so the equivalence is not unique on the nose.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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