Category

A category is a collection of objects that are linked by arrows.

Morphisms

Universal Construction

Functor

A functor is a map between categories.

Definition

Let C and D be categories. A functor F from C to D is a mapping that:

  • associates to each object \(X\) in C an object \(F(X)\) in D,
  • associates to each morphism \(f \colon X \to Y\) in C a morphism \(F(f) \colon F(X) \to F(Y)\) in D such that the following two conditions hold:
    • \(F(id_{x}) = id_{F(x)}\) for every object \(X\) in C,
    • \(F(g \circ f) = F(g) \circ F(x)\) for all morphisms \(f \colon X \to Y\) and \(g \colon Y \to Z\) in C.

Natural Transformations

A natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved.

Definition

If F and G are functors between the categories C and D, then a natural transformation \(\eta\) from F to G is a family of morphisms that satisfies two requirements.

  • The natural transformation must associate to every object \(X\) in C a morphism \(\eta_{X} \colon F(X) \to G(X)\) between objects of D. The morphism \(\eta_{X}\) is called the component of \(\eta\) at \(X\).
  • Components must be such that for every morphism \(f \colon X \to Y\) in C we have:
    \(\eta_{Y} \circ F(f) = G(f) \circ \eta_{X}\)

Monads