Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-28
How 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.
Grothendieck category
Definition
A Grothendieck category is an abelian category that satisfies AB5 and has a generator (The axioms AB5 and AB5*, Generator and cogenerator of a category).
Depends on
Used by
- The one-step generator extension functor Definition
- Abelian sheaves form a Grothendieck category Lemma
- Enough injective sheaves of modules Lemma
- Extension from subobjects of a generator detects injectivity Lemma
- Filtered colimits of abelian groups are exact Lemma
- Grothendieck abelian categories have functorial injective embeddings Theorem
- Module categories are Grothendieck categories Theorem
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, Tag 079B (standard reference, not scraped)
- Daniel Murfet, Abelian Categories, Section 2.3 (standard reference, not scraped)