M Theory Lesson 32
Any three vertex shape could represent three adicity in context and the lines that connect each vertex could be seen as a swapping morphism of sorts. I played around with this geometry a while ago and you can extend it up to n rational dimensions. A four adic system is represented by a trapezoid, rectangle, rhombus or square, each of the vertices are connected via lines and each of these lines represent swapping.
Mahndisa has been studying Matti Pitkanen's p-adic physics for quite some time now, and has a unique perspective on where all this is heading. From a categorical point of view, it is interesting to note that we are discussing new structures. I recall Batanin discussing a mysterious kind of operad which swaps sources and targets. Recall that sources and targets are idempotents for us.
Thus the usual 2-adic situation is the tip of the iceberg. And since categories can arise as algebras for operads, the p-adic logic would give rise to new kinds of category.