I don’t know about you, but all this algebraic notation starts to blur together. Wouldn’t it be nice if we could just draw pictures?
Well luckily for use we can! Just like we had diagrams for braided categories, categories with duals, and braided categories with duals, we have certain diagrammatics to help us talk about monoid objects.
First off, we think of our generating object as a point on a line. As we tensor copies of this object together, we just add more points. Then our morphisms will be diagrams in the plane. At the bottom of the diagram is the incoming object — a bunch of marked points — and at the top is the outgoing object — another bunch of marked points. In between, we have morphisms we can build from the two basic pieces we added: multiplication and unit.
See? For multiplication, two points come in. They move together and multiply, leaving one point to go out. For the unit, a point comes “out of nowhere” to leave the diagram.
As before, we set two diagrams side-by-side for the monoidal product and stack them top-to-bottom for composition. Now, what do those associativity and identity relations look like?
Neat! Associativity just means we can pull the branch in the middle to either side of the threefold multiplication, while identity means we can absorb a dangling free end.
I haven’t bothered to render a diagram for symmetry, but we can draw it by just having lines cross through each other. The naturality of the symmetry means that we can pull any morphism from one side of a crossing line to the other.
And now what about comonoid objects? We’ve got diagrams to talk about them too!
Here’s a comultiplication and a counit. We just flip the multiplication and unit upside-down to dualize them. And we do the same thing for the coassociativity and coidentity relations.
The one thing we have to take careful note of here is that everything in sight is strict. These diagrams don’t make any distinction between and ; or between , , and .