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Monad on a category

Definition

Let $C$ be a category. A monad on $C$ is a triple $(T, η, μ)$ consisting of an endofunctor $T : C \to C$ (Covariant functor, identity functor, composite functor, and contravariant functor) and natural transformations (Natural transformation and its components)

$η : 1_{C} \Rightarrow T, μ : T^{2} \Rightarrow T,$

called the unit and multiplication, such that

$μ \circ T μ = μ \circ μ T : T^{3} \Rightarrow T,$

$μ \circ T η = 1_{T} = μ \circ η T : T \Rightarrow T .$

Here $T μ$ , $μ T$ , $T η$ , and $η T$ are whiskerings in the sense of Whiskering and horizontal composition of natural transformations. Componentwise, for every object $A$ ,

$μ_{A} \circ T (μ_{A}) = μ_{A} \circ μ_{T A}, μ_{A} \circ T (η_{A}) = 1_{T A} = μ_{A} \circ η_{T A} .$

Depends on

Used by

G-sets are strictly monadic over sets Corollary
Algebra and algebra homomorphism for a monad Definition
Codensity monad Definition
Comonad on a category Definition
Distributive law between two monads Definition
Finitary functors and finitary monads Definition
Idempotent monad Definition
Morphisms between monads on one category Definition
A monoid defines the writer monad by adjoining an accumulated output Example
For a group G the monad G×(-) on sets has the G-sets as its algebras Example
The Kleisli adjunction for the maybe monad is monadic but not strictly monadic Example
The Kleisli category of the maybe monad is the category of sets and partial functions Example
FALSE: A monad is a monoid object in the endofunctor category for every category False statement
FALSE: a monoid object in an endofunctor category is the definition of a monad False statement
The monoid description of a monad requires an endofunctor category Remark
A distributive law makes the composite endofunctor a monad Theorem
A monoid object in a small endofunctor category is exactly a monad Theorem
Every adjunction induces a monad on the domain of its left adjoint Theorem
Every algebra is the coequalizer of its canonical pair of free algebras Theorem
Kleisli composition is associative and unital Theorem
On a preorder the monads are exactly the monotone extensive maps with T(Tp) below Tp; on a poset they are exactly the closure operators Theorem
Singleton and union define the covariant power-set monad Theorem
The codensity construction satisfies the monad laws Theorem

Dependency tree · two levels

5 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 5.1.1 (standard reference, not scraped)
B. Richter, From Categories to Homotopy Theory, Definition 6.1.1 (standard reference, not scraped)
S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Chapter VI.1 (standard reference, not scraped)