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.