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
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