I want to get back to discussing tangles. So far we’ve been thinking about tangles entirely topologically. But as it turns out, tangles are also fundamentally algebraic objects. The algebraic gadget we need to understand tangles is that of a free ribbon category. Indeed, Shum’s theorem states that framed, oriented tangles form the morphisms of a free ribbon category on a single generator.
To begin to understand this deep statement we must start with the definition of a category. A category is a set of objects along with a class (for technical reasons a class, not a set) of morphisms . Each morphism has a source object and a target object so that we can think of a morphism as an arrow . There is a composition operation of morphisms which is defined only if the source of is the target of . There is also an identity morphism for every object whose source and target are both . Finally we require that composition be associative and unital .
Tangles form morphisms in a category. Just let the objects be points in a plane; then clearly tangles form morphisms with their bottom endpoints as source and their top endpoints as target (or vice versa, it’s just a convention). We can compose tangles by placing them one atop the other, so long as their sources and targets match up. Identity tangles are simply a bunch of vertical lines connecting matching top and bottom endpoints. Clearly, associativity and unitality hold so tangles do indeed form a category.
We can form a category of tangles with a completely different composition however. Instead of placing tangles atop each other, we can place them side by side. Now the empty tangle is the identity. Also, in this category there is only 1 object since we can always place tangles next to each other; there’s nothing to match up! Something with 2 different categorical structures like this is called, logically enough, a 2-category. But, as we said, the second category structure has a unique object. These kinds of 2-categories are so common they get their own name, monoidal categories. Thus, tangles form the morphisms of a monoidal category.
Actually, that’s not the end of the story! We could put the tangles side by side in different ways, since the endpoints live in planes, we have 2 dimensions to work with. The two independent ways of placing tangles next to each other in addition to the standard composition of placing them atop each other turn tangles into a 3-category. Since both ways of putting tangles next to each other can be done without worrying about matching this is a special kind of 3-category called a doubly monoidal category. Doubly monoidal categories always have a way of transforming the monoidal product (side-by-side placement) into its opposite (side-by side placement but in the reverse order). This comes from the fact that the 2 monoidal structures are essentially the same. Try to think about why this is true for tangles.
Let’s think about how to transform two points sitting side by side into the same two points sitting in the opposite order. As we transform in two dimensions rotating one around the other, we trace out the familiar crossing. Of course we can rotate them in the other direction and get the other crossing.
In general, this sort of thing is called a braiding, and doubly monoidal categories always have them. For this reason, they’re also called braided monoidal categories.
Orientation means that the endpoints of our tangle are more than just points. They have directions associated with them, either up or down. We call this a dual structure, since the dual of up is down. This is familiar from linear algebra where to each vector space we can associate a dual vector space of linear maps from to the field of scalars. The important structure relating vector spaces and their duals are the evaluation and coevalutation maps. Evaluation takes a dual vector and a vector and evaluates to the scalar . Coevaluation makes use of the isomorphism where is the space of endomorphisms of . The coevaluation takes a scalar to that scalar multiple of the identity. Now, we have the same sort of structure morphisms in the category of tangles, the caps and cups. This makes the category of tangles a monoidal category with duals, just like the category of linear transformations of vector spaces.
Since cups and caps may be oriented in 2 different ways, we have 2 dual structures, a left and a right dual. The same can be said of the category of vector spaces but there, one simply identifies left and right duals. In the category of tangles it’s not so easy. Instead one must build a natural isomorphism between left and right duals and for this you need a twist. A twist is what it sounds like, take your endpoints and twist them around 360 degrees. This is where framing comes into play. If you do this to a single endpoint, you get a ribbon with a full twist in it. This has a blackboard diagram that looks like either side of the framed Reidemeister 1 move.
What if you had 2 endpoints? Think about this for a bit, you get 2 crossings between 2 ribbons each of which has a full twist in it. Luckily this is the compatibility condition between the braiding and the twist that is required of a so-called ribbon category.
To recap, a ribbon category is a braided monoidal category with duals and a twist. All of these may be defined algebraically but have intuitive topological definitions in the category of tangles. The fact that algebra may be thought about topologically can be rigorously summed up in the statement of Shum’s theorem given at the beginning of the post: framed, oriented tangles form the morphisms of a free ribbon category on a single generator.