April 22nd, 2008

(no subject)

So I think I finally have a putative definition of weak ω-category that I find satisfying:
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 → iddom(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-1f 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.

Cnm(a,b) = k (codm a *n-m0 codm b)
Dnm(a,b) = k (domm a *n-m0 domm b)

a *nm b =
• a on b (if m=n)
• idm(Cnm+1(a,b)) om (a *nm+1 b) om idm(Dnm+1(a,b)-1) (if m<n)

Where idn(x) = idid...x and similarly with domn and codn. The operation on is the "n-horizontal" composition, the one that's n steps more horizontal than pure vertical composition, and *n0 is the thing you should interpret as weak n-horizontal composition.

This stuff also lends itself well to nice pictures: