Beyond Duality

After talking to kneemo recently, I noticed that the current trend in categorical physics had not swayed far from its roots a decade or two ago. A topological field theory is still a functor from a category of spaces to a category of algebraic objects, although these may now be higher dimensional categories with a rich structure. In noncommutative geometry, one uses an algebraic description of a space to generalise spaces beyond commutative ones. But even here, spaces are still spaces and algebras are algebras.

Descartes must be rolling in his grave, to say nothing of the modern algebraic geometers. How artificial, this distinction between algebra and space. Nice functors are endofunctors, so we should really start with a category that contains all algebras and spaces. Fortunately, we might be able to make them all look like matrices! Computer scientists like Vaughn Pratt have long taken this idea seriously, and they study just the kind of dualities that are used to define noncommutative geometry.

Of course, triality should also make its appearance, if physics has its fair say. So much to do.