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.
The empty biproduct is a zero object
Statement
If the empty family has a biproduct, then that object is a zero object.
Facts & Assumptions
Given: An empty biproduct object .
A biproduct is simultaneously a coproduct and a product (Biproduct).
The empty coproduct is initial and the empty product is terminal (Limits of empty diagrams are terminal objects, and colimits of empty diagrams are initial objects).
A zero object is an object that is both initial and terminal (Initial object, terminal object, and zero object).
Proof
By [L1], the empty biproduct object is both an empty coproduct and an empty product.
By [L2], those two roles make it initial and terminal.
Therefore it is a zero object by [L3].
Depends on
Used by
Dependency tree · two levels
8 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)