Naturality

Natural Transformation

Natural Transformation

Given categories and and covariant functors , a natural transformation consists of:

• morphisms , is called the component of at .
• and components must be such that for every morphism  in  we have:

Vertical Composition

Suppose and are natural transformations between parallel functors . Then there is a natural transformation whose components Such is called the vertical composition of and .

Remark

We shall check the vertical composition is a natural transformation. Naturality of and implies that for any in the domain category , each square, and thus also the composite rectangle, commutes:

Natural Isomorphism

A natural isomorphism is a natural transformation in which every component is an isomorphism. In this case, the natural isomorphism may be depicted as

The Functor Category

Functor Category

Define the category of functors having functors as objects and natural transformations as arrows. And for each functor object , the natural transformation has components And the composite natural transformation of and is the vertical composition.

Proposition

If categories and are small, then is again a small category, but if and are locally small, then need not be. This is only guaranteed if is locally small and is small.

Lemma

A natural transformation is a natural isomorphism if and only if it forms an isomorphism in the functor category.

Lemma

Lemma Given locally small categories , , and , a map of arrows and objects, forms a functor iff

• is functorial in each argument: and are functors for all and .
• satisfies the following interchange law. Given and , .
  

Proposition

is cartesian closed, with the exponentials .

Horizontal Composition

Horizontal Composition

Given a pair of natural transformations and as indicated in the diagram:

there is a natural transformation whose component at is defined as

Lemma

Lemma Middle Four Interchange
Given functors and natural transformations

the natural transformation defined by first composing vertically and then composing horizontally equals the natural transformation defined by first composing horizontally and then composing vertically:

Proof