The n-Category Café

If you want you can try following along. Let’s try the following: All the elements you need to build any SKI combinator expression can be found in the 3-Cells. Specifically, only the first four, S, K, I and Branch will be needed. The other two occur automatically during evaluation.

Let’s build a simple test-case expression:
- Click on “Branch”. A t-junction with one end on top and two on the bottom will appear. Click on the bottom left end just inside the border of the image. Select “S Combinator”. The t-junction will now have a blue leaf. Next, click on the right branch and find an I-combinator to add a green leaf. After that, by clicking on the right side of the image, you can add another wire. Click between the two wires on the top and select branch. Click on the lose end on the bottom and add another I combinator. Click on the right end again to add another wire and at the top to connect the top two wires with another Branch.
At this point you are looking at a Mockingbird. It will repeat what ever you put into it. It’s just waiting for another bird to mock. And what better bird to mock than another Mockingbird?
Click on the bottom loose end to add a Branch. On its left end, add another Branch. And on the three loose ends, from left to right, add an S-, I- and I-combinator.

What you are looking at here is the famous infinite loop of two Mockingbirds about to mock each other:

This is already in evaluation order. When hovering over individual pieces you can confirm that: - the blue vertices are S combinators. - the green vertices are I(dentity) combinators. In this particular diagram there are no K combinators. They would be red.

• the grey vertices are branching points. You can also think of these as parentheses, sort of. They force the evaluation order and also are crucial for applying S-, K-, and I-moves. Evaluation order is depth-first from left to right, if you bring the diagram into the standard form as seen in the image. So what you are looking at in this tree is: a Mockingbird SII acting on a Mockingbird SII like so: SII(SII).

I combinators take a single input that they return unchanged. Effectively all they do is remove the branch they are attached to.
K combinators take two inputs and return the first (so they delete the second.)
S combinators take three inputs, copy the third and apply it to the first and then the second.

At this point, just to get a history of what’s happened, it might be a good idea to click on Identity or just press I on your keyboard.

So, since in the above diagram this is going to be the first move that happens, let’s look at how an S move looks like in this diagram:

First, how things need to look to apply the move:

You can see here how the move can be applied whenever an S combinator is directly next to three branches at once. These are its three inputs. Now let’s apply this move. (This can be done simply by clicking on the S-combinator. If it happens to not work, make sure your tree is arranged in precisely this form! Globular doesn’t know isomorphisms. Gotta put the work of rearranging in yourself.)

Okay, this looks a bit messy. But remember, the bottom three lines are the inputs. So just follow along upwards and see what happens. Go back and forth in time (click up and down in the second Slice field) to get a feeling for what just changed.
First, look at the right-most wire. That upside-down branch is actually a copy node. It will copy what ever is below it. Crucially, it is NOT an inverse of the Branch node and as such will NOT cancel out. It wouldn’t work if that was the case. Its right copy stays on the right side. Its left copy, however, crosses over the second input. If you remember the evaluation order that means, first the left input is evaluated, then the left copy of the right input:
Sxyz = xz(yz) - that’s exactly what’s going on here!

Furthermore you have the two remaining I combinators, but look, one of them is all tangled up! So let’s first move the left I combinator all the way (three times) up to the branch vertex, then click it. In accordance to its rule, it will have deleted its branch. Once again, if you need more time to appreciate the change, go back and forth on the Slice counter.

Next, let’s move the other I combinator up. At the crossing, drag it to the top-right. At that point it will once again be in position to be applied, so let’s do that.

At this point you get what looks kind of weird, but it really is where the magic happens. Once you have an intuitive understanding of these graphs you’ll see exactly what’s going to happen from hereon out. First, a branch and a copy node come directly together, forming a loop. You might be tempted to assume the two could cancel out, but no, the copy node only ever moves down and can’t interact at all with any vertex above it.
So when you click on it, it will actually copy the branch below it. The result is a little messy if you aren’t used to it, though not nearly as bad as the mess the S-combinator initially caused. Once again go back and forth in time and also follow the wires carefully to really see what’s happening: The copy vertex turned our one branch into two along each of which flows another copy branch. The left branch’s right copy crosses over to become the right branch’s left copy and vice versa. This way, once copy is done, there will be a perfect copy of the entire initial branch.

So let’s go on moving the left copy process along. Once again it copies a branch, spawning two more copy nodes in the process. Next you could copy the S combinator and finally finish one of the branches. After that, move the left remaining copy down twice and click it once more. Only one copy process to go but let’s untangle the the wires on the left first. Move the intersection down and the S-combinator across.
After that, let’s drag the copy node all the way down to copy. From here on out all that’s left is some untangling and you’ll soon be back to where it all began. The second mockingbird was perfectly mocked by the first which, thus, turned into yet another mockingbird. It can repeat the process all over.

Since this process didn’t feature K-nodes at all, for completeness sake I will also show you what happens with there. First, the setup when it can be applied: In the 4-cells, find the cell called “apply K”. What you get will look nothing like what the two pictures suggest. The reason for that is that the Projection level is automatically set to 2 (so 2 dimensions are dropped.) To fix that, lower it to 1.

By advancing the Slice-counter by 1 you can see what happens if you apply the move.

Technically, at the end of that right wire, there is a vertex but it is black. I thought that works well given that it symbolizes what’s happening: This wire is now dead. Every time you click on its end when right above another node, it will delete what follows below. That’s what K does: Given two inputs, it returns the first. The second one is just discarded.
If you’d rather be able to actually see the Eat vertex, you can easily change its color by finding “Eat” in the 3-Cells and click on the color field reading 000000. Pick your favorite color. I’ll ask for more feedback first but if I find this to be a common concern I’ll just upload a new version with such a change myself.

And finally, if you want to play around with this some more, say, if you want to start simulating untyped lambda calculus or the like, or if you want to recreate all the species in the book “How to Mock a Mockingbird”, there is plenty of literature available online. Just search for that title or SKI combinator calculus. It won’t take long to find some examples to try and replicate.

I added an extra rule (eating copied branches) and while I was at it I reworked the colors a bit. Do you like this better? updated version

That’s interesting, since what I had always liked about your pictures is how naturalistic and unlike Globular’s pictures they are!

very well spotted.