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So I think I finally have a putative definition of weak ω-category that I find satisfying:

A strict ω-category C

Subsets W and U of its cells

Maps α and -

A map k : {x ∈ C|x an object or (dom x ∈ W and cod x ∈ W)} → U

(x

cod(x) = dom(x

αx : x

dom(kx) = x

cod(kx) ∈ W

That's it!

W is really the underlying set of the weak ω-category, while C is kind of just scaffolding. U is a collection of "equivalence" cells in C. The conditions on α and -

The point is that the strict composition in C and the choice of representatives k can be used to define

C

D

a *

Where id

This stuff also lends itself well to nice pictures:

**Data**A strict ω-category C

Subsets W and U of its cells

Maps α and -

^{-1}, both of type U → UA map k : {x ∈ C|x an object or (dom x ∈ W and cod x ∈ W)} → U

**Conditions**(x

^{-1})^{-1}= xcod(x) = dom(x

^{-1})αx : x

^{-1}x → id_{dom(x)}dom(kx) = x

cod(kx) ∈ W

That's it!

W is really the underlying set of the weak ω-category, while C is kind of just scaffolding. U is a collection of "equivalence" cells in C. The conditions on α and -

^{-1}means that if we have a cell f : x → y in U, we also have f^{-1}: y → x in U, with ff^{-1}and f^{-1}f both being U-equivalent to identities, and so on.The point is that the strict composition in C and the choice of representatives k can be used to define

*weak*composition, whose associativity properties should fall naturally out of the higher-dimensional geometry of composition.C

_{n}^{m}(a,b) = k (cod^{m}a *_{n-m}^{0}cod^{m}b)D

_{n}^{m}(a,b) = k (dom^{m}a *_{n-m}^{0}dom^{m}b)a *

_{n}^{m}b =- a o
_{n}b*(if m=n)* - id
^{m}(C_{n}^{m+1}(a,b)) o_{m}(a *_{n}^{m+1}b) o_{m}id^{m}(D_{n}^{m+1}(a,b)^{-1})*(if m<n)*

Where id

^{n}(x) = id_{id...x}and similarly with dom^{n}and cod^{n}. The operation o_{n}is the "n-horizontal" composition, the one that's n steps more horizontal than pure vertical composition, and *_{n}^{0}is the thing you should interpret as*weak*n-horizontal composition.This stuff also lends itself well to nice pictures: