Precategories are precisely placed in the universe of neocategories. Some classical results from category theory are shown to carry over to precategories. Other results are shown not to hold in general. The natural setting for defining traversals is the class of shapely data types. We model the unit by a distinguished one-move game called Joker.
Special rules apply to the joker move.
Proofs are modelled by what we call conditionally exhausting strategies, which are deterministic and total only at positions where no joker move exists in the immediate neighbourhood, and satisfy a kind of reachability condition called P-exhaustion. We use the model to give an analysis of a counting problem in free autonomous categories which is a natural extension of the Triple Unit Problem. The first part attempts to begin the investigation of the Hilbert Q-modules. The basics of the theory of Hilbert Q-modules are established. They play a large role in noncommutative topology and geometry, being a noncommutative version of a vector bundle.