* Re: question about omega-category
@ 1998-05-20 2:46 Sjoerd Erik CRANS
0 siblings, 0 replies; 2+ messages in thread
From: Sjoerd Erik CRANS @ 1998-05-20 2:46 UTC (permalink / raw)
To: categories; +Cc: gaucher, scrans
Philippe Gaucher <gaucher@irma.u-strasbg.fr> wrote:
> Dear categorician (or categorist, I do not know the word in English),
>
>
> I posted the following question some days ago in sci.math.research.
> Maybe this list is more appropirated :
>
> I would need to understand a proof of the following proposition :
>
> There is only one functor up to isomorphism TENS : omega-Cat x
> omega-Cat -> omega-Cat for which C TENS - and - TENS C have right
> adjoint for every omega-category C and which satisfies I^p TENS I^q =
> I^{p+q} where I^p is the omega-category canonically associated to the
> p-cube, (using for example the set of composable sub pasting schemes
> of the pasting scheme associated to the p-cube).
>
> I have a paper from Crans ("Pasting schemes for the monoidal biclosed
> structure on omega-Cat") which proves explicitely the proposition. I
> do not understand the construction, which is very technical (*). Is there
> a less complicated way to prove this proposition ? I do not need an
> explicit construction.
>
> Any help is welcome.
>
> pg.
>
>
> (*) What is a pasting presentation for example ? I know the definition
> of pasting scheme, realization of pasting schemes, but I do not know the
> one of "pasting presentation". Another question : if f, g are morphisms
> in C and h, k morphisms in D, is there in C TENS D elements corresponding
> to f TENS h and g TENS k, and if the answer is yes, is it true that
> (f TENS h) o_p (g TENS k) = (f o_p g) TENS (h o_p k) ? I suppose that
> it is false : if it should be true, why the construction of this monoidal
> structure is so complicated ?
>
A pasting presentation for an omega-category is similar to a presentation
for a group, but taking into account that there are cells of different
dimensions. In particular, generators in dimension n are ``labeled''
n-dimensional pasting schemes with the labeling involving generators
*and relations* in lower dimensions. More details about pasting presentations
can be found in my paper "Pasting presentations for omega-categories", which
is available via my web-page: http://www.mpce.mq.edu.au/~scrans/papers/
or via hypatia: http://hypatia.dcs.qmw.ac.uk/author/C/CransSE/.
The proposition above follows by general categorical methods, using the
adjunction between omega-categories and cubical sets. Apart from my proof
cited above there are earlier proofs by Al-Agl and Steiner [Nerves of
multiple categories, Proc. London Math. Soc. (3) 66 (1993) 92-128] and
by Brown and Higgins [Tensor products and homotopies for omega-groupoids
and crossed complexes] (who only do omega-groupoids but their proof holds
for omega-categories as well).
Yes, there are elements in C TENS D corresponding to f TENS h and g TENS k.
But is not just that (f TENS h) o_p (g TENS k) = (f o_p g) TENS (h o_p k)
is not true: it does not even make sense, because the sources and targets
don't match up.
Sjoerd Crans
^ permalink raw reply [flat|nested] 2+ messages in thread
* question about omega-category
@ 1998-05-13 13:35 Philippe Gaucher
0 siblings, 0 replies; 2+ messages in thread
From: Philippe Gaucher @ 1998-05-13 13:35 UTC (permalink / raw)
To: categories
Dear categorician (or categorist, I do not know the word in English),
I posted the following question some days ago in sci.math.research.
Maybe this list is more appropirated :
I would need to understand a proof of the following proposition :
There is only one functor up to isomorphism TENS : omega-Cat x
omega-Cat -> omega-Cat for which C TENS - and - TENS C have right
adjoint for every omega-category C and which satisfies I^p TENS I^q =
I^{p+q} where I^p is the omega-category canonically associated to the
p-cube, (using for example the set of composable sub pasting schemes
of the pasting scheme associated to the p-cube).
I have a paper from Crans ("Pasting schemes for the monoidal biclosed
structure on omega-Cat") which proves explicitely the proposition. I
do not understand the construction, which is very technical (*). Is there
a less complicated way to prove this proposition ? I do not need an
explicit construction.
Any help is welcome.
pg.
(*) What is a pasting presentation for example ? I know the definition
of pasting scheme, realization of pasting schemes, but I do not know the
one of "pasting presentation". Another question : if f, g are morphisms
in C and h, k morphisms in D, is there in C TENS D elements corresponding
to f TENS h and g TENS k, and if the answer is yes, is it true that
(f TENS h) o_p (g TENS k) = (f o_p g) TENS (h o_p k) ? I suppose that
it is false : if it should be true, why the construction of this monoidal
structure is so complicated ?
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