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.
Eilenberg–Moore category of a monad
Definition
For a monad on , the Eilenberg–Moore category has the -algebras as objects and the -algebra homomorphisms as morphisms. Its identities and composition are those of ; Algebra homomorphisms are closed under identities and composition proves that these operations are closed and satisfy the category laws.
The Eilenberg–Moore forgetful functor sends to and acts as the identity on each underlying morphism.
Depends on
Used by
- For a group G the monad G×(-) on sets has the G-sets as its algebras Example
- FALSE: The Kleisli and Eilenberg–Moore categories are equivalent for every monad False statement
- A distributive law makes the composite endofunctor a monad Theorem
- A monad morphism induces restriction of algebras and a natural comparison of free algebras Theorem
- Algebras for an idempotent monad form a reflective subcategory Theorem
- Algebras for the covariant power-set monad are posets with all small suprema and their morphisms preserve every small supremum Theorem
- For a unital ring R, the free-R-module monad on sets has left R-modules as its Eilenberg–Moore algebras Theorem
- The Eilenberg–Moore forgetful functor creates every colimit in the base that the monad and its square preserve Theorem
- The Eilenberg–Moore forgetful functor strictly creates every limit that exists in the base Theorem
- The free-group monad has groups as its Eilenberg–Moore algebras Theorem
- The free-monoid monad has monoids as its Eilenberg–Moore algebras Theorem
- The free–forgetful Eilenberg–Moore adjunction induces the given monad Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 3 results over 3 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- E. Riehl, Category Theory in Context, 2nd ed., Definition 5.2.4 (standard reference, not scraped)
- B. Richter, From Categories to Homotopy Theory, Section 6.2 (standard reference, not scraped)