Weak omega-categories

This paper proposes to define a weak higher-dimensional category to be a simplicial set satisfying properties. The definition is a refinement of that suggested at the end of [St3] which required extra structure on the simplicial set. The paper [St3] constructed the simplicial nerve of a (strict) ω-category. The principal aim of the paper was to prove that the construction was right adjoint to the functor which assigned, to the n-simplex, the free n-category thereon (called the n-th oriental). However, I also provided emperical evidence for a precise conjecture characterizing nerves of ωcategories. Roberts [Rs] had distinguished certain simplicial sets with extra structure that he called complicial sets. Verity and I have been using the adjective stratified for a simplicial set with certain elements distinguished, and called th in, such that no elements of dimension 0 are thin and all degenerate elements are thin. For nerves of strict ω-categories, the thin elements of the nerve are the "commutative simplexes". Without actually having defined the nerve, Roberts achievement was to recognize which unique horn filler conditions might possibly characterize a stratified simplicial set as isomorphic to the nerve of an ω-category: Roberts' conjecture was that his category of complicial sets was equivalent to the category of ω-categories. In [St3] I made explicit a larger class of unique horn filler conditions and conjectured that the nerve I had constructed provided the equivalence between the category of ω-categories and a category of stratified simplicial sets satisfying these stronger conditions. I still expected that Roberts' conditions would be enough. Right at the very end of [St3] I recorded my recognition that it should be possible to define the nerve of what were coined there weak ω-categories. However, these simplicial sets should be stratified by taking the thin elements to be those simplexes which "commute up to weak equivalence". My main idea was that the same horn filler conditions should still be satisfied but without uniqueness . In [St4] I proved that the nerve of a strict ω-category as constructed in [St3] satisfied my more general unique horn filler conditions, and hence, also those of Roberts.