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Algebra and algebra homomorphism for a monad
Definition
Let be a monad on (Monad on a category). A -algebra is an object together with a morphism , called its structure map, satisfying
For -algebras and , a -algebra homomorphism is a morphism in such that
Depends on
Used by
- Algebras for a preorder monad are exactly its fixed objects up to preorder equivalence; on a poset they are its fixed points Corollary
- G-sets are strictly monadic over sets Corollary
- Free algebra for a monad Definition
- The open-set family induced by an ultrafilter algebra Definition
- 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
- A continuous map of compact Hausdorff spaces is an ultrafilter-algebra homomorphism Lemma
- The ultrafilter-limit map of a compact Hausdorff space is an algebra for the ultrafilter monad 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
- Algebra homomorphisms are closed under identities and composition 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
- Every monadic functor is conservative Theorem
- Groups are strictly monadic over sets Theorem
- Monoids and unital rings are strictly monadic over sets Theorem
- The Eilenberg–Moore forgetful functor strictly creates coequalizers of U^T-split pairs Theorem
Dependency tree · two levels
4 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 5.2.4 (standard reference, not scraped)
- B. Richter, From Categories to Homotopy Theory, Definition 6.2.1 (standard reference, not scraped)