Subject: categories: about dimension of morphisms

From: Philippe Gaucher <[email-masked]>
Date: Mon, 23 Nov 1998 16:45:58 +0100

Bonjour, 

I think that I have found a small pathological behaviour with 
dimension of morphisms in omega-categories. This is the following
one.

Let us first  recall that 
       min{m/s_m(x)=x} = min{n/t_n(x)=x} =: dim(x). 
Because x = s_m(x) = t_m s_m(x) (because m=<m) = t_m(x).

Lemma 1 : Let u and v be two composable morphisms of dimension
1 (i.e. t_0(u)=s_0(v)). Suppose that t_0(v)<>s_0(u) (<> means
different). Then u o_0 v is of dimension 1.

Proof : Suppose that u o_0 v of dimension 0. Then 
s_0(u) = s_0(u o_0 v) = u o_0 v and
t_0(v) = t_0(u o_0 v) = u o_0 v
So s_0(u) = t_0(v) : false.

Here is now the pathological behaviour : 

If s_0(u) = t_0(v), here is an example of category with u and
v of dimension 1 and u o_0 v of dimension 0 : 

We take A={alpha,beta,u,v} (four elements)

-------------------------------------
	alpha	beta	u	v
-------------------------------------
s_0	alpha	beta	alpha	beta
-------------------------------------
t_0	alpha	beta	beta	alpha
-------------------------------------

and for o_0 : 

-------------------------------------
	alpha	beta	u	v
-------------------------------------	
alpha	alpha	xxxx	u	xxxx
-------------------------------------
beta	xxxx	beta	xxxx	v
-------------------------------------
u	xxxx	u	xxxx	alpha
-------------------------------------
v	v	xxxx	beta	xxxx
-------------------------------------

(at the intersection of row x and column y, read x o_0 y)

All axioms of categories seem to be verified (?). The only composition
is a "word" of finite length "... u (beta) v (alpha) u (beta) v..."

In (A,s_0,t_0,o_0), u and v are of dimension 1 and u o_0 v
of dimension 0 because u o_0 v = alpha. 

Where am I wrong ? (observe that this behaviour cannot appear with 
for example a category coming from a composable pasting scheme).

pg.