question about weak omega category

[Tom Leinster <T.Leinster@dpmms.cam.ac.uk>]

> There is a conjecture which claims that the category 
> of cubical omega-categories is equivalent to the category 
> of globular omega-categories. If I understand correctly, 
> the conjecture was proved  in some richer framework 
> but seems to be (in my knowledge) still open as stated 
> above. 
> 
> My question is : is there a similar conjecture for 
> weak omega-category ? Is there a notion of cubical
> weak omega-category somewhere in the literature
> and a notion of globular weak omega-category ?

There is certainly a notion of globular weak omega-category: in fact, there
are at least two such. One is Batanin's, another is mine. (If you already
have an early version of the preprint of mine cited below then it will say
that the definition I present *is* Batanin's. He's since pointed out that
it's different.) I've also sketched out how one might define weak cubical
omega-category in a similar style, although there's one important hole in
this which I haven't been able to fill. There have probably been other
attempts to get a notion of weak cubical omega-category.

I think that the conjecture you describe (for *weak* omega-categories) must
be beyond our reach for a little while yet, if it's even plausible. One
reason is that we have to say what the morphisms are in the category of weak
[cubical] omega-categories. If you took the morphisms to be strict functors
(i.e. those preserving composition on the nose) then I suspect the conjecture
would fail. A more natural and plausible choice would be the weak functors
(those maps preserving composition up to coherent equivalence). However,
we seem not to understand weak functors very well at the moment. Batanin has
a definition of weak functor for the notion of weak omega-category he
presents, but I don't know that a similar thing has been done in the cubical
context. So it may not even be possible to *formulate* the conjecture in
today's language, let alone prove it.

Tom


References:

M. Batanin,
Monoidal globular categories as a natural environment for the theory of weak
$n$-categories (1997). Advances in Mathematics 136, pp. 39--103. 
Also available via 
http://www-math.mpce.mq.edu.au/~mbatanin/papers.html 

Tom Leinster,
Structures in higher-dimensional category theory (1998).
Available via http://www.dpmms.cam.ac.uk/~leinster/ 
(chapter II is the relevant bit)