Ternary Geometry III

Topological field theory enthusiasts like extending the 1-categorical constructions to the world of 2-categories. A candidate source category is then a category of spaces with boundaries which themselves have boundaries. That is, the vertices are the objects, the edges the 1-arrows and surfaces 2-arrows. In the world of ternary geometry this brings to mind the three levels of the generalised Euler characteristics, which were seen as cubed root of unity analogues to the alternating signs that occur in the world of 2. Since the boundary of a boundary is not necessarily empty, it makes more sense to look at the cubic relation $D^3 = 0$ than the usual homological $D^2 = 0$ of duality. Since the latter arises from a fundamental categorical concept, namely monads, one would like to understand the ternary categorical construction. This is why M Theory looks at ternary structures such as Loday's algebras and higher dimensional monads.