Functor Categories
If C is any category and I is a small category, we can form the functor category CI having as objects all functors from I to C and as morphisms the natural transformations between those functors. This forms a category since for any functor F there is an identity natural transformation 1F : F → F (which assigns to every object X the identity morphism on F(X)) and the composition of two natural transformations (the "vertical composition" above) is again a natural transformation.
The isomorphisms in CI are precisely the natural isomorphisms. That is, a natural transformation η : F → G is a natural isomorphism if and only if there exists a natural transformation ε : G → F such that ηε = 1G and εη = 1F.
The functor category CI is especially useful if I arises from a directed graph. For instance, if I is the category of the directed graph • → •, then CI has as objects the morphisms of C, and a morphism between φ : U → V and ψ : X → Y in CI is a pair of morphisms f : U → X and g : V → Y in C such that the "square commutes", i.e. ψ f = g φ.
More generally, one can build the 2-category Cat whose
- 0-cells (objects) are the small categories,
- 1-cells (arrows) between two objects and are the functors from to ,
- 2-cells between two 1-cells (functors) and are the natural transformations from to .
The horizontal and vertical compositions are the compositions between natural transformations described previously. A functor category is then simply a hom-category in this category (smallness issues aside).
Read more about this topic: Natural Transformation
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