**Data**

A strict ω-category C

Subsets W and U of its cells

Maps α and -

^{-1}, both of type U → U

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

**Conditions**

(x

^{-1})

^{-1}= x

cod(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: