OVERVIEW AND STRUCTUR E xv
theory of such stratified simplicial sets in chapter 1 and later, in chapter 2, we
discuss filtered semi-simplicial sets which may be used to provide an alternative
foundation to our work and which we apply at a strategically important point in
the sequel.
In chapter 6 we introduce a class of stratified sets which, for reasons that shall
become apparent, we choose to call pre-complicial sets. Pre-complicial sets satisfy a
relatively weak constraint, which allows us to make inferences about the thinness of
simplices that are related by virtue of being faces on a suitable higher dimensional
thin simplex. Our primary result in this chapter is that the full subcategory of
pre-complicial sets admits a (non-symmetric) biclosed monoidal structure, which
will become our main tool in much of what follows.
In chapter 7, we finally define Roberts' complicial sets to be pre-complicial sets
in which all admissible horns have unique thin fillers. Most of the effort in this
chapter is devoted to extending the results of chapter 6 to the reflective full sub-
category of complicial sets. In particular we show that we may reflect our biclosed
monoidal structure on pre-complicial sets down onto the category of complicial sets.
In chapter 8, we use the biclosed monoidal structure on complicial sets to build
internal and enriched categories from complicial sets. Specifically, we present a
construction which allows us to derive a category of paths in a complicial set whose
homsets are themselves complicial sets. Later we show that, in a suitable sense, we
may iterate this construction to provide an explicit equivalence inverse to Street's
nerve functor.
In chapter 9 we build analogues of the traditional simplicial decalage construc-
tion on the categories of complicial sets and complicially enriched categories. Using
an argument which deploys a certain complicial double category with connection
derived in the previous chapter, we show that these constructions correspond to
each other under the action of our path category functor. Applying simplicial re-
construction to these then establishes the fact that the path category construction
provides us with a fully faithful representation of complicial sets as complicially
enriched categories.
Finally, in chapter 10 we quickly review Street's work on parity complexes
[Str91], [Str94] and o;-categorical nerves [Str87], [Str88] and establish its con-
nection to our work on the theory of complicial sets. In particular, we construct a
canonical isomorphism between the free ^-category on the parity complex product
of a pair of simplices and the ^-category obtained by reflecting the tensor product
of corresponding standard simplices. This confirms our intuition that the compli-
cial tensor generalises the (lax) Gray tensor product of 2-categories [Gra74] and
that, under Street's nerve construction, it coincides with the ^-categorical tensor
products of Crans [Cra95] and Steiner [Ste91].
Once we have done this we can partially free ourselves from the complicial world
and discuss the all important relationship between Street's nerve construction and
our own path category construction. Finally, we use this to provide a quick and
easy proof that Street's nerve provides an equivalence between the categories of
^-categories and complicial sets, as originally conjectured in [Str87].
Dominic Verity
October 19, 2004
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