Historical Background
This work presents a proof of a result that has sometimes been referred to
as the Street-Roberts conjecture (after [Str87]). This postulates an equivalence
between the category of strict ^-categories and a category of structures called
complicial sets which are certain kinds of enhanced simplicial sets originally studied
by Roberts [Rob78].
The genesis of this work dates back to the mid-1970 and Roberts' work on
non-abelian cohomology. His original interest in this topic grew from his conviction
that (strict) (^-categories were the appropriate algebraic structures within which to
value such theories [Rob77]. This led him to define complicial sets to be simplicial
sets with distinguished elements, which he originally referred to as "neutral" then
later as "hollow" but for which we prefer the term thin (after [Dak77]), satisfying
some natural conditions related to those that characterise Kan complexes in the
homotopy theory of simplicial sets [GZ67].
In particular, his conditions include a certain kind of unique thin horn filler
condition. These closely resemble the horn fillers of Kan, with the notable excep-
tions that only certain admissible horns (for which a specified class of faces are
required to be thin) are assumed to have fillers, that those fillers are all themselves
thin and that all such fillers are assumed to be unique.
Roberts was motivated in this definition by his observation that it should be
possible to naturally generalise the classical nerve constructions of Algebraic Topol-
ogy to provide a functor from the category of (strict) ^-categories to the category
of simplicial sets which, in a suitable sense, encapsulated a natural notion of higher
non-abelian cocycle. While the nerve construction on groups and partially ordered
sets, and their common generalisation to categories, is well known and easily de-
scribed to students of Algebraic Topology, the same cannot be said of its generalisa-
tion to cj-categories. Indeed its very definition poses substantial technical challenges
which eluded Roberts at the time.
Even without an explicit construction of this nerve functor, Roberts set about
studying those simplicial sets that would occur as the nerve of some a;-category.
He observed that this functor would not provide a fully-faithful representation and
suggested rectifying this failing by introducing thin elements into his study. He
then set to characterising those thinness enhanced simplicial sets which would arise
in the replete image of the postulated fully-faithful functor and thus complicial sets
were born.
One might forgive Roberts if his identification of this category were to prove
deficient in some way, after all he was working without a nerve functor and at a time
when the theory of strict (^-categories had advanced little further than fundamental
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