# Functors

As we mentioned before, we can’t have “the set of all sets,” even though it might sound cool.

Similarly, even though it would be awesome, we can’t have the “category of all categories.” We can get away with something slightly smaller though: The category of all small categories. Let’s call it $\textsc{Cat}$. There are lots and lots of small categories. So many that in fact $\textsc{Cat}$ is a large category.

So you know what the objects are in $\textsc{Cat}$ (they are the small categories). What is a morphism in this category? If $\mathcal C$ and $\mathcal D$ are small categories, it’s something $F:\mathcal C\to\mathcal D$. We call it a functor, and it’s a way to converting one category into another.

All of this being careful business about small and large categories was just to motivate functors (and to teach you a little set theory), but in fact, we can define them for large categories as well. So here we go:

Let $\mathcal C$ and $\mathcal D$ be categories. A (covariant) functor is a map $F:\mathcal C\to\mathcal D$ in the following sense

• For each object $X\in Ob(\mathcal C)$, there is a corresponding object $F(X)\in Ob(\mathcal D)$.
• For each morphism $f:X\to Y$ in $\mathcal C$, there is a corresponding $F(f):F(X)\to F(Y)$ in $\mathcal D$.

We also have some conditions on how functors work.

• For every $X\in Ob(\mathcal C)$, Then $F(1_X)=1_{F(X)}$
• If $X\xrightarrow{f}Y\xrightarrow{g}Z$ in $\mathcal C$, then $F(g\circ f)=F(g)\circ F(f)$.

Functors probably seem scary when they’re described like this. They aren’t. They are a wonderful tool, and a beautiful construction. You see them all over the place and probably don’t even realize it. Anytime you take an object of one type, and turn it into an object of a different type in a way that “respects morphisms,” you’re applying a functor. Next time we’ll see a lot of examples of functors. Try first to come up with one or two of your own. Without seeing any examples, this should prove quite challenging, but it’s a good exercise.