User blog:Ubersketch/Formal definition for a fundamental sequence system

Define a notation as an ordered pair (O, a[n]) such that:
 * There exists 0 in the set O such that there exists no a such that S(a)=O
 * For all a in the set O there exists, S(a) such that S(a)=S(b) implies a=b and vice versa.
 * Denote all objects in O that are not S(a) where a is an object in O, or 0 as L. a[n] is a function a[n]:L*N->O such that: a[n+1] > a[n] (Note that n in a[n] is a natural number, and not all elements of L may have a[n] defined for it.)

Categorification
Along with the rules I already mentioned we can add an additional set F to the fundamental sequence.

Define a notation category as an ordered pair (O, F, a[n]) such that: Doing this turns a notation into a category and we can do normal category-theoretic things like functors and natural transformations.
 * There exists 0 in the set O such that there exists no a such that S(a)=O
 * For all a in the set O there exists, S(a) such that S(a)=S(b) implies a=b and vice versa.
 * Denote all objects in O that are not S(a) where a is an object in O, or 0 as L. a[n] is a function a[n] : L*N->O such that: a[n+1] > a[n] (Note that n in a[n] is a natural number, and not all elements of L may have a[n] defined for it.)
 * f is in the set F iff is a function f : O->O