TheoremStatement: 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.
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Homology is a homological functor on the homotopy category

Statement

Let $A$ be an abelian category. For every $n \in Z$ , the functor $H_{n} : K (A) \to A$ is homological.

Facts & Assumptions

Given: An abelian category $A$ and a standard cone triangle in $K (A)$ .

Proof

1.1given

The cone long exact sequence identifies $H_{n} (C) \to H_{n} (D) \to H_{n} (Cone (f))$ as an exact sequence, and homology factors through $K (A)$ .

2.1step 1.1given∎

By definition, a distinguished triangle is isomorphic to a standard cone triangle. Functoriality of $H_{n}$ transports the exact three-term sequence in step 1.1 across such an isomorphism, which is exactly the homological-functor condition.

Depends on

Dependency tree · 2 levels

Sources

Charles A. Weibel, Chapter 10, Lemma 10.1.4