Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07
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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.

An additive functor on abelian categories induces an exact functor on homotopy categories

Statement

An additive functor F:A→B between abelian categories induces an exact functor K(F):K(A)→K(B).

Facts & Assumptions

Given: An additive functor F between abelian categories.

Proof

1.1given

Applying F degreewise sends chain maps and homotopies to chain maps and homotopies, and preserves the finite biproduct and zero-map formula defining a cone.

2.1step 1.1given∎

Hence F(Cone⁡(f))≅Cone⁡(Ff) compatibly with shift, and it sends each standard cone triangle to one; this is the required exact-functor datum.

Depends on

Used by

Dependency tree · two levels

19 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