Subject: categories: computation in CPS

From: Philippe Gaucher <[email-masked]>
Date: Mon, 9 Nov 1998 11:06:13 +0100 (MET)

Bonjour, 

Here is a question about composable pasting schemes (CPS).

In the omega-category In generated by the n-cube, is it possible to
find a kind of "general" formula for the (n-1)-source (target) of 
the n-morphism corresponding to the interior of In ? I can do 
mechanical computation in low dimension but I am not able for the
moment to imagine a formula for any dimension. In high dimension ,
computations become very long.

For example, using notations of Crans/Johnson/Street etc..., in I2, we
have (R(x) means the CPS generated by x, sometimes also denoted by (x)) : 

s_1(00)=R(-0,0+) (almost the definition in a CPS)
and t_0(-0)=s_0(0+)=-+ => s_1(00)=R(-0) o_0 R(0+) (1) because the
union is the composition in the framework of CPS. And 
t_1(00)=R(+0,0-)=R(0-) o_o R(+0) (2). Obvious with a picture.

In I3, we have : 

s_2(000)=R(-00,0+0,00-)=R(-00,0++,-0-,0+0,00-,++0)
t_0(-00)=s_0(0++) => R(-00,0++) = R(-00) o_0 R(0++)
t_0(-0-)=s_0(0+0) => R(-0-,0+0) = R(-0-) o_0 R(0+0)
t_0(00-)=s_0(++0) => R(00-,++0) = R(00-) o_0 R(++0)

and 

t_1(R(-00) o_0 R(0++)) = t_1(-00) o_0 R(0++) (axiom of omegaCat)
                       = (-0-) o_0 (-+0) o_0 (0++) with (2)
                       
s_1(R(-0-) o_0 R(0+0)) = R(-0-) o_0 s_1(0+0) (axiom of omegaCat)
                       = (-0-) o_0 (-+0) o_0 (0++) with (1)
                       
=> R(-00,0++,-0-,0+0) = R(-00,0++) o_1 R(-0-,0+0)

and in the same way, we preove that 
t_1(R(-0-,0+0))=s_1(R(00-,++0))  

=> s_2(000) =((-00) o_0 (0++)) o_1 ((-0-) o_0 (0+0)) o_1 ((00-) o_0 (++0))

For t_2(000), read the above formula from the right to the left and 
replace - by +. Almost obvious with a picture.

For I4 now : 

I have found a formula for s_3(0000)... A little bit long and not
interesting. 

For I5 : Too long. 

More generally, the question is : for a CPS, is there a way to compute 
the source and target of a R({x}) using only compositions of elements 
like R({y}) ?

Thanks in advance for your help.

pg.