From: Philippe Gaucher <gaucher@irma.u-strasbg.fr>
To: categories@mta.ca
Subject: computation in CPS
Date: Mon, 9 Nov 1998 11:06:13 +0100 (MET) [thread overview]
Message-ID: <199811091006.LAA04175@irma.u-strasbg.fr> (raw)
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.
next reply other threads:[~1998-11-09 10:06 UTC|newest]
Thread overview: 3+ messages / expand[flat|nested] mbox.gz Atom feed top
1998-11-09 10:06 Philippe Gaucher [this message]
1998-11-09 21:31 Sjoerd CRANS
1998-11-11 22:21 Ross Street
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