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 ring as a one-object Ab-enriched category
Example
Every ring determines a one-object -enriched category whose single hom-object is the additive group of and whose enriched composition is multiplication in .
Facts & Assumptions
Given: A ring .
A one-object preadditive category is the same thing as a ring (A one-object preadditive category is the same thing as a ring).
A preadditive category is exactly an -enriched category (Ab-enriched categories are exactly preadditive categories).
Verification
By [L1], is the endomorphism ring of a one-object preadditive category.
Applying [L2] to that preadditive category turns it into a one-object -enriched category, with the additive group of as hom-object and ring multiplication as enriched composition.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
9 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 (standard reference, not scraped)