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.
A preadditive category with a zero object has zero morphisms in the published sense
Statement
Every preadditive category with a zero object has a canonical system of zero morphisms in the sense of the published zero-morphism definition, and those morphisms are the zero elements of the hom-groups.
Facts & Assumptions
Given: A preadditive category with a zero object .
A zero object supplies a unique compatible system of zero morphisms (A zero object supplies a unique compatible system of zero morphisms).
In a preadditive category with a zero object, that system agrees with the hom-group identities (In a preadditive category with a zero object, the zero morphism is the neutral element of each hom-group).
In a preadditive category, initial and terminal coincide (In a preadditive category, an object is initial exactly when it is terminal).
Proof
Since is a zero object, [L1] gives a compatible zero-morphism family in the published sense.
By [L2], for every pair of objects these same morphisms are the additive identities of the hom-groups. Thus the preadditive structure and the published zero-morphism structure coincide.
So a preadditive category with a zero object has zero morphisms in the published sense. The role of [L3] is exactly to make the term "zero object" stable inside the preadditive setting.
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
- The Stacks Project, Section 12.3: Preadditive and additive categories (standard reference, not scraped)