Ivancevic_Applied-Diff-Geom

Technical Preliminaries: Tensors, Actions and Functors 129to higher categorical structures [Leinster (2002); Baez (1997)].A category C with only one object is a monoid (= semigroup with unit)M. A 2−category C with only one 0−cell is a monoidal category M. Abraided monoidal category is a monoidal category equipped with a mapcalled braidingA ⊗ B β A,B ✲ B ⊗ A ,for each pair A, B of objects.The canonical example of a braided monoidal category is BR [Leinster(2003)]. This has:(1) Objects: natural numbers 0, 1, . . .;(2) Morphisms: braids, e.g.,(taken up to deformation);there are no morphisms m ✲ n when m ≠ n;(3) Tensor product: placing side–by–side (which on objects means addition);and(4) Braiding: right over left, e.g.,Knots, links and braids are all special cases of tangles (see [Reshetikhinand Turaev (1990)]). The mysterious relationships between topology, algebraand physics amount in large part to the existence of interesting functorsfrom various topologically defined categories to Hilbert, the categoryof Hilbert spaces. These topologically defined categories are always∗−categories, and the really interesting functors from them to Hilbert arealways ∗−functors, which preserve the ∗−structure. Physically, the ∗ operationcorresponds to reversing the direction of time. For example, there isa ∗−category whose objects are collections of points and whose morphisms