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 . If set-theoretical questions make you nervous, read this as “small categories”. It does work out for general categories, though. We know that is cartesian, and thus monoidal. We can take pairwise products of categories, and the terminal category is — 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 enriched over ? We’ve got a collection of objects, and for each pair of objects in we have a category of morphisms. In each of these we have a collection of objects (called “1-morphisms”), and for each pair of 1-morphisms in we have a collection of morphisms (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 to and one from to and gives a composite 1-morphism from to . Since it’s a functor, it also acts on 2-morphisms. If is a 2-morphism in (that is, both and go from to ) and is a 2-morphism in , then we get a composite 2-morphism . Of course, we also can take and and get a composite 2-morphism by using the composition in the category . The composition functor is associative.
For each object there’s an identity 1-morphism . And then it has an identity 2-morphism . The 1-morphism acts as the identity for the composition functor , and it’s easy enough to verify that is not only the identity for the composition in , but it’s also the identity for the composition 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 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: . 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 with one object? Well, we have our object , and a category . Any two 1-morphisms (the objects of this category) can be composed with each other by , 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”, instead of , and instead of . 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.