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.
Freyd's axioms A0, A1, A1*, A2, A2*, A3, and A3* for abelian categories
Definition
Freyd's axiomatisation of an abelian category asks for the following data and properties.
- A0. A zero object exists (Initial object, terminal object, and zero object).
- A1. Every pair of objects has a product.
- A1*. Every pair of objects has a coproduct.
- A2. Every morphism has a kernel (Equalizers and coequalizers as limits and colimits of a parallel pair).
- A2*. Every morphism has a cokernel (Equalizers and coequalizers as limits and colimits of a parallel pair).
- A3. Every monomorphism is a kernel (Monomorphism and epimorphism by left and right cancellation).
- A3*. Every epimorphism is a cokernel.
Unlike the working definition on this page, no additive enrichment is part of the data. Freyd's point is that the additive structure can be recovered from the remaining axioms.
Depends on
Used by
Dependency tree · two levels
6 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
- Junhan Tan, The Freyd-Mitchell Embedding Theorem, Definition 2.2 (standard reference, not scraped)
- Peter Freyd, Abelian Categories, Chapter 2 (standard reference, not scraped)