M Theory Lesson 215

Category theory is very good at generalising the concept of topological space. Dropping the assumption that membership of a point in an open set is a two valued entity leads to the idea of a Chu space. Vaughan Pratt runs this amazing website, complete with a Chu space calculator. A space is represented by a rectangular array with rows indexing points and columns indexing dual points, called states. The entries usually come from a simple set.

The basic two truth values {0,1} cover a lot of ground. For example, a topological space has points and open sets as dual points. The entries from {0,1} determine whether or not a point belongs to an open set. Using a table of 8 (= $2^{3}$) truth values, one can describe the category of groups. The category of all Chu spaces, with respect to a given truth value set, has an interesting self dual structure.

In M theory we also write down matrices with entries restricted to small sets. Moreover, the 0 and 1 appearing in the two dimensional spin operators do have an interpretation as Boolean truth values, since the category of finite sets is almost the same thing as a category of finite dimensional Hilbert spaces.