The primary purpose of this work is to characterise strict (^-categories as
simplicial sets with structure. We prove the Street-Roberts conjecture in the form
formulated by Ross Street in his work on Orientals, which states that they are
exactly the "complicial sets" defined and named by John Roberts in his handwritten
notes of that title (circa 1978).
On the way we substantially develop Roberts' theory of complicial sets itself
and make contributions to Street's theory of parity complexes. In particular, we
study a new monoidal closed structure on the category of complicial sets which
we show to be the appropriate generalisation of the (lax) Gray tensor product of
2-categories to this context. Under Street's u;-categorical nerve construction, which
we show to be an equivalence, this tensor product coincides with those of Steiner,
Crans and others.
While this the Street-Roberts conjecture is of interest in itself, its real impor-
tance lies in the fact that it acts as a prelude to the development of a simplicial
rendition of the theory of weak ^-categories, known as weak complicial set theory.
Indeed much of the work on (lax) Gray tensor product of complicial sets given here
routinely generalises to this broader context. This allows us to enrich the category
of weak complicial sets over itself in a natural way and opens the possibility of
providing a coherence result for Street's weak cj-categories along the lines of the
tricategorical coherence result of Gordon, Power and Street.
This work has been designed to be as self contained as possible, and conse-
quently much of the material covered in its first five chapters is classical in nature.
It has been reprised 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 interest that the
field of higher category theory has garnered from mathematicians and theoretical
physicists of diverse backgrounds, it was felt prudent that no prerequisite assump-
tions be made. Most particularly, it was recognised that some readers might not be
fully conversant with certain of the more abstract aspects of the general theories of
enriched and internal categories or with the 2-dimensional specifics of the theories
of 2-categories and double categories.
Received by the editor February 1, 2005.
2000 Mathematics Subject Classification. Primary 18D05, 55U10; Secondary 18D15, 18D20,
18D35, 18F99, 18G30.
Key words and phrases, simplicial set, cj-category, parity complex, oriental, simplicial nerve,
horn filler, Kan complex, double category, Gray tensor product, decalage.