## March 24th, 2008

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I seem to have settled into a rather late sleep schedule. I don't much care for it, actually. Anyway.

Today involved enjoyable meetings-up at coffee tree roasters and later meatloafings-up with . Ground beef + ground turkey + cheese makes good meatloaf. I think it could have stood even more cheese, but I'm not complaining. Later nathan and mark and her and I played "probe", a weird old hangman-like game from the 60s. Fun, though.

I had some interesting further thoughts about formulating a certain kind of higher-dimensional graph based on an intuition that a pasting diagram of n-cells looks a lot like an n-sphere missing one cell. That is, you can express the (n-1)-dimensional domain and codomain of such an "n-path" by an extra cell that "wraps around" the composition into a cycle so that now every (n-1)-cell has exactly one n-cell to whose domain it belongs, and exactly one n-cell to whose codomain it belongs. While spacing out at the 7 party last night I thought I had discovered a fatal flaw in this plan, but now I think I see how to work around it.

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Here is my current intuition for the geometric setup I am trying to use to think about weak ω-categories:

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.