ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Homology as a homological delta functor

Example

Fix an abelian category $A$ . For chain complexes in $A$ that vanish in negative degrees, the family $H_{n} : {Ch}_{\geq 0} (A) \to A, n \geq 0,$ together with the usual connecting morphisms of homology, is a homological delta functor.

Facts & Assumptions

Given: An abelian category $A$ and its abelian category of nonnegatively graded chain complexes.

[L1]

This page's concrete homology family is the candidate homological delta functor (The homological delta-functor carried by homology of complexes).

[L2]

Homology of complexes satisfies the exactness and naturality axioms (Homology of complexes satisfies the delta-functor naturality and exactness laws).

[L3]

A homological delta functor is exactly such a family of additive functors with natural connecting maps (Homological delta functor).

Verification

technique · direct

1.1L1given

Restrict the integer-indexed homology family of [L1] to complexes that vanish in negative degrees. This full subcategory is closed under kernels and cokernels degreewise, hence is abelian, and its short exact sequences have $H_{- 1} = 0$ .

2.1L2L3step 1.1∎

The exactness and naturality axioms required in [L3] are supplied by [L2]; step 1.1 makes the integer-indexed long sequence terminate as $H_{0} (C) \to 0$ . Therefore the restricted family is a homological delta functor indexed by $n \geq 0$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 `Derived Functors` (standard reference, not scraped)