HISTORICAL BACKGROUND xi
and became convinced that Street's parenthetical remarks regarding a potential
simplicial definition held great merit. In ignorance of [Str88], I set about proving
that in any nerve Street's admissible horns had unique fillers and in a very short
time, and to my great pleasure, I succeeded in constructing a decomposition of each
oriental which established this result.
Spurred on by my initial success, I decided to turn my hand to proving Street's
version of Roberts' original conjecture and soon succeeded in showing that Street's
nerve functor was fully faithful. While this result in itself was a clear contribution
to the then extant state of knowledge in this area, I was not satisfied since the
method used to do so had originally been conceived as a proof of the complete
Street-Roberts conjecture. My initial reaction to this block was to cast around
for a categorical abstraction which might strengthen my result, but to no avail.
Somewhat disheartened by this, I wrote notes on my proof to date and circulated
this handwritten manuscript to a few interested parties (including Street) before
settling down to complete my PhD work on enriched category theory.
Returning to this problem in 1993, I realised that an argument based upon the
decalage construction would close the gap in my proof. In 1994 I gave talks on the
resulting proof to the Sydney (now Australian) Category Seminar, at the University
College of North Wales in Bangor, at the Mathematical Sciences Research Institute
in Berkley and to a Peripatetic Seminar on Sheaves and Logic at the Newton
Institute in Cambridge UK.
Unfortunately, however, career events overtook me before I had time to commit
the final proof to paper. In early 1995 I followed up a series of consultancy engage-
ments in the financial markets by accepting a full time role in investment banking.
I spent the subsequent 5 years as a derivative securities analyst and trader manager
before returning to academia in mid 2000.
On returning to this field, two things struck me immediately. Firstly, and
most pleasingly, an area of study that had given so much joy to a small band of
enthusiasts had grown into a dynamic area of wide debate and interest, driven by
an influx of new ideas and approaches. Secondly, and much to my relief, nobody
appeared to have provided a proof of the Street-Roberts result upon which I had
devoted so much time prior to my sojourn into the business world.
The proof presented herein is fundamentally no different to the one that I spoke
on in 1994. The primary innovation on that original work has been the adoption
of a (lax) Gray tensor product of complicial sets as a unifying organisational and
I should like to dedicate this work to three groups of people. Firstly of course to
my wife, Sally, and children, Lottie and Florrie, who have put up with a husband
and father incessantly crouched over the word processor. They have never once
questioned the importance to me of completing this work and have sustained me in
body and soul over the 12 years it has been in gestation. Secondly I would like to
thank Ross Street, who has been the most formative influence in my development
as a mathematician. It was his friendship, support and inspiration which convinced
me to return to academe and to work which I had long since convinced myself I
would never commit to paper. In the end it was his quiet reminders, to the effect
that I owed it to the community to write up my ideas, that spurred me to find the
time in a busy schedule to write these scratchings. Finally I should like to thank
the staff of the Postgraduate Professional Development Program of the Division of