Dear David On 27/09/2010, at 11:07 PM, David Leduc wrote:

I've never heard before the term "co-omega-category". What is it?

...

since mathematicians traditionally like to describe constructions in terms of elements rather than co-elements,

I'd love to hear to co- side of the story if you have it in store.

To answer this I have to be a bit more specific about what I mean by an omega-category. Here is one way to define it. Start with an omega-graph A (also called globular set); that is, a family of sets A_n indexed by the natural numbers n ≥ 0 together with functions s_n , t_n : A_{n+1} --> A_n satisfying the globular identities (s_n) (s_{n+1}) = (s_n) (t_{n+1}) and (t_n) (s_{n+1}) = (t_n) (t_{n +1}); so in fact by composing consecutive s and consecutive t we have a directed graph s , t : A_n --> A_m for all 0 ≤ m < n. An omega-category is an omega-graph A equipped with a category structure on each of the graphs s , t : A_n --> A_m such that each of the 2-graphs A_n --> A_m --> A_p becomes a 2-category for 0 ≤ p < m < n using these category structures. This structure on the graph involves composition functions from the pullback of s , t : A_n --> A_m to A_n and identity equipment functions A_m --> A_n satisfying axioms which can be stated as commutative diagrams of functions. From this description we can see that we can define omega-categories in any category X with pullbacks; just replace "set" and "function" above by "object of X" and "morphism of X". A co-omega-category in X is an omega-category in X^op. In particular, there is a co-omega-graph in omega-Cat involving the family of omega-categories 2_n and the co-source and co-target omega-functors 2_n --> 2_{n+1} which pick out the two (non-identity) n-cells in 2_{n+1}. The co-compositions are defined on pushouts rather than pullbacks. Notice that 2_0 is the ordinal 1 and 2_1 is the ordinal 2. You should get the idea if I point out that the pushout of the two different functors 1 --> 2 is the ordinal 3; and the co-composition 2 --> 3 takes 0, 1 in 2 to 0, 2 in 3. Iff A is a co-omega-category in X then each X(A,C) is an omega- category naturally in C. Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]