M Theory Lesson 27
In M theory we would also like to consider higher dimensional operads. For quaternionic number theory it is appropriate to start with 3-sphere discs, plus further structure. Rather than marked boundary points, for instance, a 3-sphere can contain knots and links and also surface boundary components. If the surface pieces are punctured spheres they can be made to look like 2-disc diagrams. We could pack enormous amounts of algebraic information into such an operadic structure. Batanin's 2-level tree composition is a guide to horizontal and vertical compositions in the 2-operad case.