xii PREFAC E
Information and Communication Sciences at Macquarie University, of which I am
the academic director. Without their support and hard work, filling in for me while
I wrote this work, I would never have been able to find the silence and space to
organise these ideas.
Relationships to Other Work
While this work predominantly interests itself in a study of strict u;-categorical
structures, it is nevertheless squarely motivated by a broader program to define
and study weakened ^-categorical notions. In particular, its purpose from the very
start has been to act as a first step toward providing a full account of the theory
of Street's simplicial weak cj-category notion, which first appeared in sketch form
in his orientals paper [Str87] and upon which he later elaborated in [Str03].
Indeed, much of the work on (lax) Gray tensor product of complicial sets given
here routinely generalises to Street's category of weak complicial sets. In particular,
using these generalisations we may show that the category of weak complicial sets
supports biclosed structures generalising those on the category of bicategories and
homomorphisms as discussed in [Str80]. These allow us to enrich the category of
weak complicial sets over itself in natural ways and opens the possibility of providing
a coherence result for Street's weak ^-categories along the lines of that described
for tricategories in [GPS95].
Notice however that, just as in the bicategorical and tricategorical cases, the
corresponding "monoidal structure" on weakly complicial sets is only weakly adjoint
(in some suitable sense) to this biclosed structure. As a result it is only weakly
coherent, making it somewhat inconvenient to calculate with. One convenience
the weak complicial approach provides, however, is that we can always perform
all required calculations in a bigger category of pre-complicial sets, on which the
corresponding tensor is actually part of a genuine monoidal biclosed structure.
The reader will, of course, be aware that over the past ten years a wide va-
riety of weak ^-categorical notions have been proposed by various authors. No-
table amongst these are those of Joyal [Joy97], Batanin [Bat98] and Baez and
Dolan [BD98], although more may be found in the literature or in Leinster's survey
of the current state of the definitional art [Lei02]. Many of these drew their initial
inspiration from Street's parenthetical remarks in [Lei02] although none of them,
except for the summary given in Leinster [Lei02] and Street's own account [Str03],
expand directly on the purely simplicial approach analysed here. This work starts
from a point of view neatly summed up by Street's comment in [Str03] that a
simplicial formulation bears the distinct advantage that:
Simplicial sets are lovely objects about which algebraic topologists
know a lot. If something is described as a simplicial set, it is ready
to be absorbed into topology. Or, in other words, no matter which
definition of weak UJ-category eventually becomes dominant, it will
be valuable to know its simplicial nerve.
In many ways, our work here also parallels work of Ronald Brown and his
coworkers, which established analogous results for the somewhat simpler groupoidal
case. While our work and theirs share at least one common motivation, to develop
a coherent and complete theory of higher dimensional cohomology, their primary
interest is not the development of an encompassing theory of weak ^-categories
but rather the explication of a theory within which to make explicit calculations of
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