In this post, we provide the minimal category theory background that is required to define a monad.

Definition of a Category

A category $\mathcal{C}$ contains the following:

1. Objects: A sequence of objects, denoted by $obj(\mathcal{C})$: $A, B, C$
2. Arrows: A sequence of arrows denoted by $f, g, h$

The above satisfy the following properties:

1. For each arrow $f$, there exist objects $dom(f)$ and $cod(f)$ called the domain and codomain of $f$, respectively. We write $f: A \to B$ to indicate $dom(f)=A$ and $cod(f)=B$.
2. For any two arrows $f: A \to B$ and $g: B \to C$, there is an arrow $g \circ f: A \to C$ called the composite of $f$ and $g$.
3. For each object $A$, there exist the identity arrow $1_A : A \to A$
4. Associativity: $h \circ (g \circ f) = (h \circ g) \circ f$
5. Unit: $f \circ 1_A = 1_B \circ f$ for every arrow $f: A \to B$

Functor

A homomorphism of categories is called a functor. More precisely, we call $F: \mathcal{C} \to \mathcal{D}$ between two categories $\mathcal{C}$ and $\mathcal{D}$, a functor iff:

1. Arrow preservation: $F(f: A \to B) := F(f): F(A) \to F(B)$
2. Identity preservation: $F(1_A) := 1_{F(A)}$ for every $A\in\mathcal{C}$
3. Composition preservation: $F(f \circ g) := F(f) \circ F(g)$

That is, $F$ preserves domains and codomains, identity arrows, and composition.

Natural transformations

Let functors $F,G: \mathcal{C} \to \mathcal{D}$.

A natural transformation $\eta$ from $F$ to $G$ is an assignment to every object $X\in{obj(\mathcal{C})}$ of a morphism $\eta_X : F(X) \to G(X)$ in $\mathcal{D}$ such that for every morphism $f : X \to Y$ in $\mathcal{C}$ we have:

$\eta_Y \circ F(f) = G(f) \circ \eta_X$ in $\mathcal{D}$

Monad

If $\mathcal{C}$ is a category, a monad on $\mathcal{C}$ is defined as an endfunctor $T: \mathcal{C} \to \mathcal{C}$ equipped with two natural transformations:

$$\eta : 1_{\mathcal{C}} \to T$$ $$\mu: T^2 \to T$$

The above natural transformations must satisfy: (a) $\mu \circ T\mu =\mu \circ \mu T$ and (b) $\mu \circ T\eta =\mu \circ \eta T = 1_{T}$

References

1. Category Theory by Steve Adowey