M Theory Lesson 6
Points take many guises. Eventually we will have the option of throwing them away altogether, using topos theory. After all, the question that really needs answering is, what is an observable? But things get pretty interesting before then. In Brannen's Clifford algebra one hunts out idempotents, which satisfy the simple projector relation tt=t. Similarly, one might use points in projective (twistor) space, described by the Jordan algebra of Michael Rios.
Are there other ways in which points are naturally associated to operators satisfying tt=t? Why yes, using categories in a fairly simple way, as follows. One can think of the objects of a category as the identity arrows on those objects. For example, a set being a discrete category which only has objects, only has identity arrows. Now let t be the target map, sending an arrow f to its target identity arrow tf. Then clearly tt=t. Similarly, one can discuss a source map s.
For those who are really keen on playing, remember that a topology t in the form of an arrow from Omega to Omega in an elementary topos also obeys the relation tt=t.