Suppose you have a family of sets Rn indexed by natural numbers, some operations domn, codn : Rn+1 → PRn (where P indicates powerset) and a predicate vn : Rn → bool. Elements of Rn are called "n-cells" and things satisfying vn are called "virtual n-cells" — they serve to record domain and codomain in a slick way.
A subset C of Rn is an n-precycle if it has exactly one virtual n-cell in it. We write that cell as k(C) and define C* to be C without k(C). (C,D) ∈ PRn+1 × PRn+1 is a prespan if C and D are both precycles with k(C) = k(D).
Let an n-precycle P be given. A subset C of Rn+1 is balanced over P if the union of (domn x)* for x ∈ C is equal to P*, and no overlap occurs while computing this union, and so too for cod.
A 0-precycle is a 0-cycle if its cardinality is two. (Think of the two-point 0-sphere) An (n+1)-precycle P is an n-cycle if there is an n-precycle Q such that
- P is a precycle over Q
- P has no strict subset that is also a precycle balanced over Q
- Q is a union of n-cycles
- dom(k(P)) and cod(k(P)) are subsets of Q
(C,D) ∈ PRn+1 × PRn+1 is a span if C and D are both cycles with k(C) = k(D).
(Rn, dom, cod, v) is a ω-graph if (dom x, cod x) is a span for every x, and for every span (D, C), there is exactly one virtual cell x such that dom x = D and cod x = C. As well there must be exactly one virtual cell in R0.