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.
Abelian groups, modules, and vector spaces are additive
Example
The categories , , and are additive. Their hom-sets are abelian groups under pointwise addition, and their finite biproducts are the usual direct sums.
Facts & Assumptions
Given: A ring and a field .
An additive category is a preadditive category with finite biproducts (Additive category).
Left -modules and their homomorphisms form a category (Left modules over a fixed ring and module homomorphisms form the large locally small category ).
-vector spaces and linear maps form a category (Vector spaces over a fixed field and linear maps form the large locally small category ).
Verification
In each of the categories named in the Example, morphisms add pointwise, so every hom-set is an abelian group. For this is the special case of [L2], and [L3] gives the vector-space version.
Finite direct sums of abelian groups, modules, or vector spaces carry the usual injections and projections, which satisfy the product and coproduct universal properties. Therefore each category has finite biproducts. By [L1], all three categories are additive.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
15 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
- Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)