Functor categories

The establishment of natural transformations between functors will allow us to formalize all sorts of concepts and informal definitions. First, let's see how they allow us to think of functors as objects worthy of study in and of themselves.

Definition of a functor category

One can verify that $D^C$ does indeed satisfy the axioms of a category. The fact that functors between two categories have a notion of arrows between them can be seen as a hint towards some inevitable "higher category theory." A conventional category (i.e., the type we've been studying) consists of objects and arrows, but nothing beyond that. When attempting to turn the lens of category theory on itself, however, we have discovered that the "category of categories" contains objects (categories), arrows (functors between categories) and arrows between the previous arrows (natural transformations between functors).

It's hard not to see this as cracking open Pandora's Box. If natural transformations are indeed "natural," and they appear to act something like "two-dimensional arrows", then surely there is some extension of the notion of "category" to a structure that contains objects, 1-arrows and 2-arrows. How about 3-arrows? 4-arrows? Why stop?

Uh oh. We're sliding down the slope into the world of $\mathrm{\infty }$-category theory! Let's not go there, at least for now.

For now, let's simply take this observation as at least partial justification for calling arrows between categories "functors" and not "morphisms", since in most familiar situations the morphisms between your objects don't have any (obvious) notion of maps between them.

Examples of functor categories

The main examples most people stumble across are probably hom-functors (in the context of Yoneda's Lemma) and presheaves (in algebraic geometry), but for now we will satisfy ourselves with some much simpler examples.

Functors from the categories $\mathbf{\text{0}}$

Recall that the empty category $\mathbf{\text{0}}$ has no objects and no arrows. For each category $C$ there is a unique functor $\mathbf{\text{0}}\to C$, namely the empty functor (with empty object map and empty arrow map). It is straightforward to verify we have an equivalence of categories $C^{\mathbf{\text{0}}}\simeq \mathbf{\text{1}}.$ On objects, this equivalence sends the empty functor $\mathbf{0}\to C$ to the unique object of $\mathbf{1}$; on arrows, this equivalence sends the unique natural transformation from empty functor to itself (i.e., the empty natural transformation) to the unique (identity) arrow in $\mathbf{1}$.

Functors from the categories $\mathbf{\text{1}}$

Similar to the previous example, functors $\mathbf{\text{1}}\to C$ are in bijection with objects of $C$. Moreover, the natural transformations between such functors are in bijection with arrows in $C$. You should now be able to verify we have an equivalence of categories $C^{\mathbf{\text{1}}}\simeq C$.

Are you feeling a bit better about the exponential notation for functor categories?

Functors from a discrete category

Generalizing the previous two examples, let $S$ be a set and $\mathbf{S}$ be the corresponding discrete category. Then each functor $F:\mathbf{S}\to C$ can viewed as a family $\{F(s){\}}_{s\in S}$ of objects in $C$ indexed by $S$. A natural transformation $\tau :F\Rightarrow G$ is exactly a family of arrows ${\tau }_s:F(s)\to G(s)$ in $C$. (Since $\mathbf{S}$ is discrete, there is no real naturality condition to satisfy.)

So, the functor category $C^{\mathbf{S}}$ consists of families of objects in $C$ (indexed by $S$), together with arrows between such families.

Commutative diagrams of a fixed shape

Suppose $J$ is a fixed category. Recall that for any category $C$, functors $F:J\to C$ can be thought of as "commutative diagrams in $C$ of shape $J$." For example, suppose $J$ is the category with three objects and two nonidentity arrows, as illustrated below:

Then each functor $F:J\to C$ corresponds to a diagram in $C$ of the form

This particular functor category is useful when studying pullbacks and pushforwards.

Suggested next notes

Adjoints
Universal Properties III - Yoneda's Lemma