Alphabeta Math
PropositionStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-09-01‡ rests on unproved material (inherited)
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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Horseshoe resolutions are compatible with morphisms of short exact sequences up to homotopy

Statement

Assume the Axiom of Dependent Choice.

Fix a morphism between two short exact sequences, chosen projective resolutions of the two left objects and the two right objects, and chosen horseshoe middle resolutions for the two middle objects. Then any two middle comparison maps that, together with fixed side comparison maps, form morphisms of short exact sequences of complexes are chain-homotopic. Hence compatibility of chosen horseshoe middle resolutions with the induced middle morphism is only defined up to homotopy.

Facts & Assumptions

Given: A morphism between two short exact sequences, fixed side comparison maps on the chosen end resolutions, and two middle comparison maps that make the corresponding ladders commute.

[L2]

Comparison maps lifting the same morphism are unique up to chain homotopy (Projective comparison maps are unique up to chain homotopy).

[L3]

A morphism of short exact sequences of complexes is the ambient compatibility notion (A morphism of short exact sequences of complexes).

Proof

technique · direct
1.1

By hypothesis and [L3], the two chosen middle maps are comparison maps between the same pair of projective horseshoe resolutions, they lift the same middle-object morphism, and together with the fixed side maps they define morphisms of short exact sequences of complexes.

L3givenalgebra
2.1

Any two such middle comparison maps lifting the same object morphism are chain-homotopic by [L2]. Therefore the horseshoe construction is compatible with morphisms only up to homotopy, not canonically on the nose.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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