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

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.

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

Statement

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

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

Depends on

Dependency tree · 2 levels

Sources

The Stacks Project, Derived Categories, Section 13.10