M Theory Lesson 54

Returning to lattice theory, recall that the order relation between objects is represented by an arrow $a \rightarrow b$. Thus the lattice of two elements (0 and 1) is the same as the category 2.

Now in the topos Set, truth and falsity arise as follows. Truth is a given arrow from the one point set into {0,1}. Falseness is a characterising arrow in a basic pullback square, such that the square source is the initial object in the category, namely the empty set. In other words, both true and false are arrows into a special object $\Omega$. The action of turning true into false introduces the concept of complementation. Grothendieck's great insight into topological spaces was to see that $\neg \neg U$ need not be $U$, and that this was a logical statement.

In replacing binary logic by ternary logic, it is natural to assume three arrows into an object $\Omega$. This suggests the replacement of pullback squares by cubes. Instead of a lattice $0 \rightarrow 1$ we begin with the next simplex, a triangle on the objects 0, 1 and 2. The triangle has a face, as do the square faces of a cube. Complementation must be replaced by a three-way swapping of basic truth values. This approach to 3-logos theory offers a new way of generalising topological spaces to higher categorical dimensions.

Observe that the faces (2-arrows) contribute to the weaker concept of complementation. In the trivalent corner of a spatial cube they play the role of dual time. It should not be surprising, therefore, to find that mass generation is linked to the sixth face of the tricategorical parity cube, the faces of which represent internalised edges of the Mac Lane pentagon.