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.
Generator and cogenerator of a category
Definition
An object of a category is a generator when the singleton set is separating in the sense of Separating and coseparating sets of objects.
Dually, an object is a cogenerator when is coseparating.
Depends on
Used by
- A cocomplete locally small abelian category with a generator supplies the category-side SAFT hypotheses dually Corollary
- Grothendieck category Definition
- FALSE: a generator is automatically projective False statement
- FALSE: every abelian category has a generator False statement
- Generator, separator, and the three inequivalent-looking definitions Remark
- A generator detects comparison of subobjects Theorem
- A projective generator detects isomorphisms Theorem
- An AB3 locally small abelian category with a generator is well-powered Theorem
- The cancellation and epimorphism descriptions of a generator agree Theorem
Dependency tree · two levels
3 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
- Alexandre Grothendieck, Sur quelques points d'algèbre homologique, Barr translation, Section 1.9 (standard reference, not scraped)
- The Stacks Project, Tag 079B (standard reference, not scraped)