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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
- G-sets are strictly monadic over sets Corollary
- For a group G the monad G×(-) on sets has the G-sets as its algebras Example
- The comparison functor for the free-group adjunction Example
- The Kleisli adjunction for the maybe monad is monadic but not strictly monadic Example
- βℕ as the free ultrafilter algebra Example
- FALSE: The Kleisli and Eilenberg–Moore categories are equivalent for every monad False statement
- Supplied created canonical presentations give a quasi-inverse to the comparison functor Lemma
- Under dependent choice, algebras for a finitary monad on a complete cocomplete locally small category have coequalizers Lemma
- 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
- Every algebra is the coequalizer of its canonical pair of free algebras 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 coequalizers of U^T-split pairs 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 · two levels
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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)