At 22 minutes and 7 seconds into the lecture, Brendan holds up a Rubik's cube and says it's a category with one object. And then he demonstrates several rotations and says they are the arrows of the category.

But if rotations are morphisms, then wouldn't the objects of the category be the different configurations the cube could be in (in which case there's more than one object)?

That seems a more natural way to think about it at first (though maybe that's just a programmer's intuitions from graph theory sneaking in), and I think all the rules for a category (associativity, etc) are satisfied that way.

On the other hand, maybe since you have the exact same transformations available to you at each step, then a more native category theory way of thinking about it is to say it's just one object, and each arrow takes you back to that object.

Curious whether this is obvious to others.