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.
Free algebra for a monad
Definition
For a monad and an object , the free -algebra on is
It is a -algebra: its associativity axiom is the componentwise monad associativity equation, and its unit axiom is . For a morphism , naturality of makes an algebra homomorphism.
Depends on
Used by
- βℕ as the free ultrafilter algebra Example
- FALSE: Every algebra for a monad is free False statement
- Over a cocomplete base, a monadic category is cocomplete exactly when it has coequalizers Proposition
- A monad morphism induces restriction of algebras and a natural comparison of free algebras Theorem
- Every algebra is the coequalizer of its canonical pair of free algebras Theorem
- The comparison from the Kleisli category is fully faithful with image the free algebras Theorem
- The free–forgetful Eilenberg–Moore adjunction induces the given monad Theorem
Dependency tree · two levels
2 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 5.2.8 (standard reference, not scraped)
- B. Richter, From Categories to Homotopy Theory, Example 6.2.3 (standard reference, not scraped)