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.
Finitary functors and finitary monads
Definition
A functor is finitary when it preserves every small filtered colimit (Filtered categories and filtered colimits, Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors).
A monad is finitary when its underlying endofunctor is finitary (Monad on a category). No preservation claim is imposed on the unit or multiplication separately.
Depends on
Used by
- Categories of models for algebraic theories Definition
- Under dependent choice, algebras for a finitary monad on a complete cocomplete locally small category have coequalizers Lemma
- Under dependent choice, a finitary monad on a complete cocomplete locally small category has complete and cocomplete algebras Theorem
Dependency tree · two levels
10 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
- E. Riehl, Category Theory in Context, 2nd ed., Definition 5.5.4 (standard reference, not scraped)