In the second session of our category-theory club for school children we took a giant leap from the informality of diagrams towards formal mathematical notations. Our starting point were the cardboard-and-string diagrams from the previous session which we have already established that they were the same even though the strings had different lengths and colours and the bits of cardboard slightly different sizes. The reason they were the same (I chose not to use “equal” because of possibly overloaded meaning to the children) was because they were created following the same instructions.
The next step was now to draw on paper these diagrams and each child was requested to do that, and there were relatively few problems in performing this task reasonably. As one would expect, the differences between the resulting diagrams drawn by each child were now quite staggering but the idea that they were still the same diagram thankfully persisted because these were obviously accurate (as much as possible) renderings on paper of things that were the same in reality.
With this part of the work concluded successfully we moved on to what I thought will be the most challenging step: introducing a combinator notation for diagrams, inspired by compact closed categories. As a preliminary step they were required to draw the diagrams using only straight lines and right angles (the example was chosen carefully so no braiding was needed) and then we went over the diagram and tried to identify the simplest components. I actually introduced the term “component” as “a thing out of which we make other things” and after some ooh-ing and aah-ing the children seemed happy enough to accept the Big Word. Unfortunately at this critical stage a logistical problem struck: the room did not have a whiteboard or a blackboard, just an electronic “smart”-board. The irony of the term is obvious, and I am preparing a separate rant for the faddish abandonment of perfectly good classical tools of learning for fickle and fragile technology-heavy alternatives. So instead of drawing a big diagram on a big whiteboard and identifying the components I had to walk around each table and do that with each group of children, which wasted a lot of time.
The initial set of combinators we identified were: I (a line, the identity), E (a feedback loop, the compact-closed unit), U (a feedforward loop, the compact-closed co-unit) and f (a rectangle box, the only morphism so far in our category). We also discovered two ways to combine components, writing them next to each other (composition) or stacked on top of each other (tensor).
The obvious two tasks were first to write the formula corresponding to our running example diagram then drawing a diagram from a given formula. Some typical results are below.