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PREFACE

definitions. It is therefore a great tribute to Roberts' insight that our main theorem

here (theorem 266 of chapter 10) establishes his sought-for equivalence under a

definition of complicial set which only differs from his in manner of expression. He

provided a precise conjecture in 1978, our only contribution has been to prove it as

stated!

The next big advance in this story came with Street's papers on orientals

[Str87] and parity complexes [Str91]. He was introduced to this study, during

a visit Roberts made to Sydney in the (southern hemisphere) summer of 1977-8,

and was captivated by it. While he was quickly able to establish the sought for

equivalence at dimension 2, it became apparent to him that establishing the gen-

eral equivalence would be quite a difficult problem. Consequently he decided to

concentrate on providing a rigorous definition of the nerve of an u;-category.

After a couple of false starts to that end, he soon realised that the crux of

the matter was to define the free u;-category On on an n-simplex. However, in

order to make this insight precise it was necessary to define exactly the kinds

of combinatorial structures that might give rise to free ^-categories and even to

explicate the sense in which freeness itself should be interpreted. Furthermore,

once these concepts had been defined it would still be necessary to define exactly

how an n-simplex might be considered to be such a structure.

The first of these questions elicited the introduction of certain kinds of induc-

tively defined combinatorial structures called uo-computads, which provide the de-

sired definitions but are often inconvenient to calculate with. To facilitate the work

we are engaged in here, Street later introduced a restricted form of u;-computad,

called a parity complex. These satisfy some very strong loop-freeness conditions

designed to ensure that we may describe the cells of the associated free u;-category

as (pairs of) subsets of the parity complex itself.

To understand how one might render an n-simplex as a parity complex, Street

started by observing that On should play the part of some form of non-abelian

n-cocycle in the sought after cohomology theory. This led him to conjecture that

we could take insight from abelian cohomology and build a consistently oriented

parity complex whose elements were the faces of a standard n-simplex, each of

which should be oriented as a cell to "map from odd numbered faces to even num-

bered ones". Under this definition, he was able to demonstrate that the resulting

structure satisfied the strong loop-freeness conditions required of a parity complex

and thereby provide a completely explicit description of the free o;-category On.

Street completed the construction of his u;-categorical nerve by enriching his

orientals to a functor from the category of simplicial operators A and applying

Kan's construction [Kan58]. His paper [Str87] goes on to lift this functor to map

^-categories to simplicial sets with thin elements (which we call stratified sets in the

sequel) and to formalise Roberts' original conjecture in this context. He identifies a

family of admissible horns which strictly contains the class introduced by Roberts

and in [Str88] he demonstrates that within the nerve of an ^-category such horns

do indeed have unique thin fillers. It is worth noting, however, that while Street's

complicial set definition appears, at first sight, to be stronger than Roberts', our

proof here demonstrates that they are in fact equipotent.

My own contribution to this story began in 1991 when I first read [Str87]

and immediately became captivated by this problem. In particular, I had been

searching for an approach to defining structures which we now call weak uo-categories