Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-06‡ 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.

The Yoneda product agrees with derived Ext composition

Statement

Under the higher-Yoneda/derived-Ext identification, the splice of α∈YExt⁡p(L,N) and γ∈YExt⁡q(M,L) corresponds to the derived Ext composition α∘γ∈Ext⁡p+q(M,N).

Facts & Assumptions

Given: Composable Yoneda extension classes satisfying the hypotheses for the higher comparison.

Proof

technique · direct
1.1

Use the projective-resolution comparison in Higher Yoneda Ext agrees with derived Ext. A p-extension and a q-extension are represented by their lifted cocycles on the corresponding projective syzygies. Splicing the extensions concatenates their exact tails; lifting through that concatenated tail is the composite of the two lifted comparison maps.

givenconstruct
2.1

Therefore the cocycle assigned to the splice is the chain-level composite representing the derived Ext composition. Passing to cohomology identifies the splice with α∘γ. Degree zero agrees as well because both products use ordinary morphism composition, with the units fixed in The Yoneda product is associative and unital.

step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

14 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