RE: characterisation of nerve of omega-categories

["Dominic Verity" <domv@ics.mq.edu.au>]

Hi Philippe,

The answer to your question is that, to my knowledge, the characterisation
of the nerves of n-categories currently only exists in conjectural form in
the literature. In particular, a full description of this conjecture is
given in the paper:

"The Algebra of Oriented Simplexes" by Ross Street (JPAA 49 (1987) pp
283-335)

and an associated note "Fillers for Nerves" (I forget the precise reference)
which proves the necessity of this characterisation.

This actual conjecture is originally due to John Roberts - it does involve
simplicial sets enriched with a distinguished set of "hollow" or "thin"
simplices and appropriate "admissible" horn filler conditions with respect
to these thin simplices. Roberts calls these structures "complicial sets".

I presented a proof of this conjecture to a conference at UC Berkeley (MSRI)
in 1993 (I think) and also in a number of seminars given at Bangor in Wales
and at the Sydney Category Seminar, but unfortunately never published the
result, due to a subsequent career change (I became an investment banker).
Roberts' original conjecture as described by Street does indeed hold - in
fact a slightly weaker result may be proved which only involves fillers for
"inner" horns.

More recently, I have taken some time away from the world of finance and am
currently working on writing up my results in this area - which I hope to
make available over the next few months.

All the very best

Dominic Verity
Macquarie University
Sydney, Australia




> -----Original Message-----
> From: cat-dist@mta.ca [mailto:cat-dist@mta.ca]On Behalf Of Philippe
> Gaucher
> Sent: Saturday, 23 September 2000 2:07
> To: categories@mta.ca
> Subject: categories: characterisation of nerve of omega-categories
>
>
> Dear all,
>
> I don't remember where I could find a characterization for
> a simplicial set to be the simplicial nerve of some strict
> globular omega-category ? I think that  the characterization
> is that the simplicial set must be given with a structure
> of thin elements satisfying some axioms like the filling of
> horners and thin horners. Could you send me a reference please ?
>
> Thank you in advance. pg.
>
>
>