# The Unapologetic Mathematician

## 2-Categories

Here’s another example of an enriched category. This one is extremely important, and to a certain extent it’s been my goal in my coverage of category theory. It’s also a very basic chunk of what they talk about over at the n-Category Café.

The monoidal category we use is $\mathbf{Cat}$. If set-theoretical questions make you nervous, read this as “small categories”. It does work out for general categories, though. We know that $\mathbf{Cat}$ is cartesian, and thus monoidal. We can take pairwise products of categories, and the terminal category is $\mathbf{1}$ — the category with one object and one (identity) morphism. The “underlying” functor gives the collection of objects in a given category.

Okay, so what’s a category $\mathcal{C}$ enriched over $\mathbf{Cat}$? We’ve got a collection of objects, and for each pair of objects $(A,B)$ in $\mathcal{C}$ we have a category $\hom_\mathcal{C}(A,B)$ of morphisms. In each of these we have a collection of objects (called “1-morphisms”), and for each pair $(f,g)$ of 1-morphisms in $\hom_\mathcal{C}(A,B)$ we have a collection of morphisms $\hom_{\hom_\mathcal{C}(A,B)}(f,g)$ (called “2-morphisms”).

Wow, that looks confusing. Okay, let’s say it again a little differently. We have:

• a collection of objects (“0-morphisms”)
• collections of 1-morphisms that go from one object to another
• collections of 2-morphisms that go from one 1-morphism between a pair of objects to another 1-morphism between the same pair of objects

There’s also a “composition” functor between the categories of 1-morphisms. This takes a 1-morphism from $A$ to $B$ and one from $B$ to $C$ and gives a composite 1-morphism from $A$ to $C$. Since it’s a functor, it also acts on 2-morphisms. If $\phi:f\rightarrow g$ is a 2-morphism in $\hom_\mathcal{C}(A,B)$ (that is, both $f$ and $g$ go from $A$ to $B$) and $\xi:h\rightarrow k$ is a 2-morphism in $\hom_\mathcal{C}(B,C)$, then we get a composite 2-morphism $\xi\circ\phi:h\circ f\rightarrow k\circ g$. Of course, we also can take $\phi:f\rightarrow g$ and $\xi:g\rightarrow h$ and get a composite 2-morphism $\xi\cdot\phi:f\rightarrow h$ by using the composition in the category $\hom\mathcal{C}(A,B)$. The composition functor $\circ$ is associative.

For each object $C$ there’s an identity 1-morphism $1_C\in\hom_\mathcal{C}(C,C)$. And then it has an identity 2-morphism $1_{1_C}:1_C\rightarrow1_C$. The 1-morphism $1_C$ acts as the identity for the composition functor $\circ$, and it’s easy enough to verify that $1_{1_C}$ is not only the identity for the composition $\cdot$ in $\hom_\mathcal{C}(C,C)$, but it’s also the identity for the composition $\circ$ of 2-morphisms.

We call this structure a “2-category”, or more specifically a “strict 2-category”. We’ll get to weak ones eventually.

So do we know any good examples? Sure. The first is $\mathbf{Cat}$ itself! Here the objects are categories, the 1-morphisms are functors between categories, and the 2-morphisms are natural transformations between functors. In fact we already saw right when we defined a natural transformation that given a pair of categories we have a category of functors between them, which is halfway to having a 2-category right there! And then we know we have both compositions of 2-morphisms because those are just the “horizonatal” and “vertical” compositions we first needed when we talked about units and counits of adjunctions.

Speaking of adjunctions, they give another 2-category: $\mathbf{Adj}$. The objects here again are categories, but now the 1-morphisms are adjunctions between categories. And then we have conjugate pairs between adjunctions, with the “horizontal” and “vertical” compositions between them as our 2-morphisms.

And as a last example, what’s a 2-category $\mathcal{M}$ with one object? Well, we have our object $\mathbf{*}$, and a category $\hom_\mathcal{M}(\mathbf{*},\mathbf{*})$. Any two 1-morphisms (the objects of this category) can be composed with each other by $\circ$, and there’s an identity 1-morphism. Now let’s just shift our language and say “object” instead of “1-morphism”, “morphism” instead of “2-morphism”, $\otimes$ instead of $\circ$, and $\circ$ instead of $\cdot$. What we’re left with is exactly the definition of a strict monoidal category! That is: just as a category with one object is a monoid, so a 2-category with one object is a monoidal category!

There are a lot of 2-categories out there, and we’ll be mentioning many more as the time goes on.

August 16, 2007 Posted by | Category theory | 12 Comments