Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-31
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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.

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FALSE: the homotopy category is obtained by identifying quasi-isomorphisms with identities

Statement

The homotopy category is obtained by identifying quasi-isomorphisms with identities.

Facts & Assumptions

Given: The zero map from the acyclic noncontractible complex 0→Z→2Z→ mod 2Z/2→0 to the zero complex.

[A1]

The statement refuted is: the homotopy category is obtained by identifying quasi-isomorphisms with identities.

[L1]

The homotopy category keeps the same objects and uses homotopy classes of chain maps as morphisms (The homotopy category of chain complexes).

[L2]

A quasi-isomorphism is defined by its effect on homology (Quasi-isomorphism).

Refutation

technique · direct
1.1

The displayed zero map is a quasi-isomorphism, because both complexes are acyclic. Yet it is not invertible by any chain homotopy inverse, since an inverse would force the source complex to be homotopy equivalent to zero and hence contractible, which it is not.

L2givenalgebra
2.1

Therefore merely passing to homotopy classes, as in [L1], does not turn every quasi-isomorphism into an identity or even into an isomorphism. So [A1] is false: that later localization is not the definition of K(A).

A1L1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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