Let’s pick up with the diagrams for monoid objects from yesterday. In fact, let’s draw the multiplication and unit diagrams again, but this time let’s make the lines really thick.
Now we’re looking at something more like a region of the plane than a curve. We really don’t need all that inside part, so let’s rub it out and just leave the outline. Of course, whenever we go from a blob to its outline we like to remember where the blob was. We do this by marking a direction on the outline so if we walk in that direction the blob would be on our left. Those of you who have taken multivariable calculus probably have a vague recollection of this sort of thing. Don’t worry, though — we’re not doing calculus here.
Okay, now the outline diagrams look like this:
That’s odd. These diagrams look an awful lot like Temperley-Lieb diagrams. And in fact they are! In fact, we get a functor from to that sends to . That is, a downward-oriented strand next to an upward-oriented strand makes a monoid object on !
But to be sure of this, we need to check that the associativity and identity relations hold. Here’s associativity:
Well that’s pretty straightforward. It’s just sliding the arcs around in the plane. How about the identity relations?
The right identity relation holds because of one of the “zig-zag” relations for duals, and the left identity relation holds because of the other!
Now you should be able to find a comonoid object in in a very similar way.