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.

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.