# 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 structure-preserving 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 morphism F(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