Natural transformations

The trinity of concepts of category theory: categories, functors, and natural transformations form an ordered triple. So it makes sense to introduce functors before dealing with natural transformations. The introduction of functors suggested a basic perspective in which they are treated like functions, and that in turn suggested certain things about the structure and parts of functors.

Natural transformations, however, force us to change our perspective on functors. Natural transformations treat functors not like functions, but rather like indexed families of objects and morphisms. This leads to a slight change of perspective on the nature of functors, which will be dealt with now.

Morphisms of morphisms:
Let $C$ be a category, then $C^{\to}$ is its arrow category consisting of all morphisms of $C$ as objects and as morphisms ordered pairs $(i,o)$ of morphisms of $C$ that satisfy the commutative diagram: If the topos $Sets$ is the quintessential example of a category, then the topos of functions $Sets^{\to}$ is the quintessential example of an arrow category.

Functors as indexed families:
By turning the category of locally small categories into a concrete category, we considered functors as functions. What if a functor $F : C \to D$ is actually a family of objects indexed by a category $C$: objects in the category $D$ and the arrow category $D^{\to}$.

• For each $X \in Ob(C)$ we have an object $F(X)$ in $D$
• For each $m \in Arrows(C)$ we have an object $F(m)$ in $D^{\to}$

It would then seem that because functors are indexed families of objects, they can be treated as objects of a category themselves. This is indeed possible with functor categories and natural transformations, which operate on the objects indexed by a functor componentwise.

Natural transformations:
As functors are indexed families of objects, morphisms of functors are naturally morphisms of each object indexed by them. Let $F,G : C \to D$ be two functors and let $t : F \to G$ be a componentwise morphism of functors. Then $t$ maps categorical elements in $F$ to those of $G$. In particular,

• $t$ associates to each object $X$ in $C$ a $D$ morphism from $F(X)$ to $G(X)$
• $t$ associates to each morphism $m$ in $C$ a $D^{\to}$ morphism from $F(m)$ to $G(m)$

The standard way to define a componentwise morphism of functors is by defining the object mapping by a functon $t: Ob(C) \to Arrows(D)$. Then the morphism mapping is defined by doubling up the object mapping so that for any $C$ morphism $f: A \to B$ then $(t_A,t_B)$ is the morphism of morphisms $F(f) \to G(f)$. This simply means that $t$ satisfies the following commutative diagram: The componentwise nature of natural transformations explains why everything in a functor category is defined by componentwise versions of constructions in certain underlying categories. For example, in topoi of set-valued functors all limits/colimits are defined componentwise.

The componentwise nature of functor categories, suggests that the best way to understand them is by their output category $D$ and its arrow category $D^{\to}$. In particular, the most useful tools for studying any topos of set-valued functors are $Sets$ and $Sets^{\to}$ and all other constructions are defined over them. Morphisms in $D$ and $D^{\to}$ are the basic units that make up all natural transformations.

Enriched categories:
Let $Ab$ be the category of commutative groups. Then componentwise addition in $Ab$ gives each hom class $Hom(G,H)$ of commutative groups the structure of a commutative group. In general, the idea of an enriched category is that we can take hom classes of any given category, and define certain additional componentwise algebraic operations on it.

The description of natural transformations simply makes the concrete category $Cat$ into an enriched category, so that each hom class $Hom(F,G)$ has the structure of a category. These hom classes $Hom(F,G)$ are then all the different categories of functors. This suggests natural transformations are no more rare of a concept then any componentwise operation on homomorphisms encountered in abstract algebra.

New perspective on functors:
Our previous analysis of functors focused on their relationship to functions, determined by making the category $Cat$ into a concrete category. Then parts of functions could be determined on the function-level, but the idea of making functors into families means that they have different types of parts.

In the same way that a family of sets not only has subsets of itself, but also subsets of its members a functor not only has parts but it also has parts of its elements. Thus a subfunctor is defined by a natural transformation that takes each categorical element of a functor to a subobject, and a quotient functor does the same for quotients. We thus arrive at a different kind of construction on functors defined componentwise.