Alphabeta MathHow 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.
Additivity can be derived rather than postulated, depending on the axiomatisation
The working definition on this page starts from additivity because it is the cleanest form for later citations. Freyd's axiomatisation goes the other way: the additive structure is a theorem recovered from normality, conormality, products, coproducts, kernels, and cokernels.
That is why items Abelian category, Freyd's axioms A0, A1, A1*, A2, A2*, A3, and A3* for abelian categories, and Freyd's axioms force the additive structure and recover the AB2 definition coexist rather than compete. The first is the library's working interface; the second and third explain why that interface could have been packaged differently without changing the mathematics.
The uniqueness part of the recovered enrichment is already abstracted in The uniqueness of the enrichment is an Eckmann-Hilton phenomenon: once the finite biproduct law exists, there is no second compatible addition to choose.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
13 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
- Peter Freyd, Abelian Categories, Appendix (standard reference, not scraped)
- Barry Mitchell, Theory of Categories, Proposition 18.4 (standard reference, not scraped)