The Unapologetic Mathematician

Mathematics for the interested outsider


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