What is a Monad?
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:
- Objects: A sequence of objects, denoted by \(obj(\mathcal{C})\): \(A, B, C\)
- Arrows: A sequence of arrows denoted by \(f, g, h\)
The above satisfy the following properties:
- 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 \(cod(f)=B\).
- 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 \(g\).
- For each object \(A\), there exist the identity arrow \(1_A : A \to A\)
- Associativity: \(h \circ (g \circ f) = (h \circ g) \circ f\)
- 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:
- Arrow preservation: \(F(f: A \to B) := F(f): F(A) \to F(B)\)
- Identity preservation: \(F(1_A) := 1_{F(A)}\) for every \(A\in\mathcal{C}\)
- 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) \(\mu \circ T\eta =\mu \circ \eta T = 1_{T}\)