OVERVIEW AND STRUCTUR E xiii
homotopic invariants. To that end a study of groupoids rather than categories is
adequate for most purposes.
The complicial sets discussed here are most closely related to their simplicial
T-complexes, which were first discussed in Dakin's thesis [Dak77], where the term
"thin" was first coined, and later developed and popularised in the work of Brown,
Higgins and others. Since this work was groupoid oriented, the definitions involved
are somewhat simpler than those discussed here and a proof of the equivalence
between simplicial T-complexes and c^-groupoids is far more easily attained. Indeed,
a proof of this equivalence in the rank 2 case dates to Dakin's thesis in 1977.
While the simplicial approach is important to their work, it should be pointed
out that in general their preference has been to work in a cubical context. This
allows them to more easily build generalised van Kampen Theorems, for comput-
ing homotopical Invariants, and to define monoidal closed structures for analysing
homotopy classes of maps. A comprehensive and up-to-date account of this work
may be found in Brown's excellent survey article [Bro04]. In this cubical pro-
gram the closest result to the one presented here is that of Al-Agl, Brown and
Steiner [AABS02] which establishes an equivalence between the category of (strict)
^-categories and a category of multiple (cubical) categories with connection.
While this cubical approach is attractive for the reasons discussed above, it
is our conviction that it does not provide quite such a convenient context within
which to develop a theory of weak ^-categories. Nevertheless cubical calculations,
and most particularly those involved in a theory of tensor products, remain vital
even in the simplicial context. The theory discussed here, therefore, appears to
combine the best of both worlds by allowing for a theory with all of the beauty,
elegance and economy of the simplicial approach while at the same time providing
a simple and highly explicit description of the cubical Gray tensor product.
Overview and Structure
This work is almost exclusively devoted to proving the Street-Roberts con-
jecture, in the form presented in [Str87]. On the way we substantially develop
Roberts' theory of complicial sets [Rob77] itself and make some contributions to
Street's theory of parity complexes [Str91]. In particular, we study a new monoidal
closed structure on the category of complicial sets which we show to be the appro-
priate generalisation of the (lax) Gray tensor product of 2-categories to this con-
text. Under the equivalence conjectured by Street [Str87] and Roberts [Rob77],
which we prove here, this tensor product coincides with those of Crans [Cra95],
Steiner [Ste91] and others.
From the outset, it has been designed to be as self contained as possible. While
much of the material covered in its first five chapters is classical in nature, it has
been presented here in order to fix our notation for the sequel and to aggregate
together a number of familiar (and not so familiar) pieces of Algebraic Topology and
Category Theory from a diverse range of sources. Given the influx into this field of
mathematicians from diverse backgrounds, it was felt prudent that no prerequisite
assumptions be made. Most particularly, it was recognised that some readers might
not be fully conversant with certain of the more abstract aspects of general (enriched
and internal) category theory and higher category theory.
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