Breaking down "A monad is just a monoid in the category of endofunctors"
March 11, 2019
“…just a monoid…”
Associativity is a property of some binary operations where the order of operations does not matter, e.g. since addition is associative,(a + b) + c = a + (b + c)
A finitary operation is an operation with finite arity
An algebraic structure is a set A together with a collection of finitary operations on A
A semigroup is an algebraic structure with a single associative binary operation.
An identity element is an element of a set that leaves other elements unchanged when combined with them (with respect to some binary operation).
A monoid is a semigroup with an identity element
∴ monoids make sense
“…in the category…”
A morphism is a mapping from some source object A to some target object B
An identity morphism is a morphism such that the source and target objects are the same object
A category is a collection of objects linked by morphisms, where the morphisms are such that they are associative and there exists an identity morphism for each object
∴ categories make sense
“…of endofunctors”
A functor F is a structurepreserving map between categories. That is, given two categories C and D:

it associates each object X in c to an object F(X) in D

it associates each morphism
g: X → Y
in C to a morphismF(g): F(X) → F(Y)
in D such that$F(id_X) = id_{F(X)}, \space \forall X \in C$
$F(g \circ h) = F(g) \circ F(h), \space \forall g:X \rightarrow Y \in C, \space \forall h: Y \rightarrow Z \in C$
An endofunctor is a functor that maps a category to the same category
∴ endofunctors make sense
So what’s a monad?
…I’ve got no idea.