XIV

PREFACE

In particular, where categorical abstractions are concerned we assume little

more than that the reader should have a general grounding in basic categorical con-

cepts, such as functor, natural transformation, limit, colimit, comma category and

2-category etcetera, all of which may be gleaned from Mac Lane's book [Mac71].

Congruent with this philosophy, chapters 1 to 4 are all contextual in nature,

providing fairly standard presentations of traditional material. Chapter 1 consists

of a brief introduction to the theory of simplicial sets, up to and including the

theory of shuffles. While this is not intended to provide an exhaustive treatment

of the simplicial algebra necessary to read this work, it should provide most of the

necessary background and adequate pointers to the available literature.

At some points in the sequel our arguments are substantially simplified by

couching them in more abstract categorical terms, along the lines described in

Kelly's book [Kel82]. In particular, our construction of the monoidal biclosed Gray

tensor structure on complicial sets relies on Day's reflection theorem for monoidal

biclosed categories [Day70] and many of our later constructions and calculations

are couched in terms of left exact theories and their coalgebras. While a thorough

reading of Kelly's book would handsomely repay the effort involved, the results of

greatest interest here are collected together in chapter 2; the reader should refer to

the cited literature for detailed proofs.

Chapter 3 rehearses the basic definitions in the theory of (internal) categories,

double categories and cj-categories. We also remind the reader of the relationship

between double categories and 2-categories, by discussing Spencer's recognition

principle for those double categories that arise as the double categories of squares

in some 2-category (see Brown and Mosa [BM99]).

Chapter 4, the last of these contextual introductions, provides a account of the

classical simplicial decalage construction and the method of simplicial reconstruc-

tion. We review those parts of the theory of (co) monads which were developed in

order to provide a general way of constructing functors into categories of simplicial

structures. In this context, it is a classical result that the category of simplicial

sets supports a canonical comonad, called the decalage comonad, which is in some

sense generic for this construction. Again, this material will be very familiar to

Algebraic Topologists and Category Theorists but may be less familiar to others

and, indeed, the 2-categorical presentation we give here may be considered to be

somewhat non-standard.

Prom chapter 5 we concentrate on developing the theory of complicial sets and

for much of this work we study these as novel structures in their own right. Only

much later, in chapter 10, do we "tie the knot" by relating our constructions back

to the traditional theory of (strict) (^-categories. In order to do so we contribute

to the theory of parity complexes and provide a deepened analysis of Street's nerve

functor.

One of the attractions of complicial sets as a foundation for ^-category theory,

of both the strong and weak variety, is that they build upon the familiar theory

of simplicial sets. However it is too much to hope that simplicial sets themselves

are enough, especially since Street's canonical nerve construction does not provide

us with a full representation of ^-categories as simplicial sets. The issue here is

that this nerve does not record enough information about the identities in our UJ-

categories, a deficiency we rectify by storing this missing data using a structure

dubbed hollowness by Street but later renamed stratification. We examine the