M Theory Lesson 216

As explained in a nice paper by Eva Schlaepfer, the Chu construction takes a suitable (symmetric monoidal closed) category $V$ containing a truth object $K$ and constructs another category from it, which has a structure ($\ast$-autonomous) closer to applications in physics. For example, the category of finite dimensional vector spaces is of this type. The objects of the new category are cospan diagrams in $V$ of the form In the example of topological spaces, arising from the monoidal category Set, $K$ is the two point set. Usually one assumes that $V$ has pullbacks, so a cospan diagram may always be completed to form a square. If $A$ is the terminal one point set, such a pullback square would be a classifying square for the topos Set. An arrow in the new category is specified by a commuting square with target $K$ for a source span diagram $(f \otimes 1 , 1 \otimes g)$, where $(f,g): (A,B) \rightarrow (A',B')$ is a pair of arrows between the objects in $V$.

The unit for $\otimes$ in the new category is $(I,K)$, where $I$ is the unit in $V$. In other words, if finite dimensional vector spaces over $\mathbb{C}$ arose in this way, the unit one dimensional space would be specified by a pair $(\bullet,\mathbb{C})$ in Set. Observe how this resembles the quantum mechanical notion of state. From the logos point of view, thinking of $\mathbb{C}$ as a truth set which is much larger than {0,1} illustrates the logical complexity of complex spaces.