Alphabeta Math
PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-01‡ rests on unproved material (inherited)
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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Homology of complexes satisfies the delta-functor naturality and exactness laws

Statement

For every abelian category, the family of homology functors together with the connecting morphisms of this page is a homological δ-functor: it sends each short exact sequence of complexes to a long exact sequence, and it is natural under morphisms of short exact sequences.

Facts & Assumptions

Given: An abelian category and a short exact sequence of complexes in it.

[L1]

This page defines the family (Hn,∂n) as a concrete homological δ-functor candidate (The homological delta-functor carried by homology of complexes).

[L2]

Short exact sequences of complexes carry long exact homology sequences (The long exact sequence in homology).

[L3]

Those long exact sequences are natural under morphisms of short exact sequences (The long exact homology sequence is natural).

Proof

technique · direct
1.1

By [L1], the only axioms left to check are exactness for each short exact sequence and naturality for each morphism of such sequences.

L1given
2.1

Exactness is exactly [L2], and naturality is exactly [L3]. Therefore the family of homology functors with these connecting morphisms satisfies the required δ-functor laws.

L2L3step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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