Australian Category Seminar

This talk presented an alternative description of Street's orientals. The description is geometric. The elements of the orientals turn out to be certain simplicial sets. A finite n-dimensional simplicial set A is called compatible if for x, y $\in$ X_n, x $\partial_i$; = y $\partial_j$, i+j even, implies x=y. A compatible simplicial set is "sensibly oriented" at its top dimension. To analyze its lower dimensions we define the domain and codomain of a simplicial set. If A is an n-dimensional simplicial set, x $\in$ A is called an end of A if there exists y $\in A_n$, {a_0, a_1, ... a_k } even integers a_0 < a_1 ... < a_k with x = y $\partial_{a_k} \partial_{a_{k - 1}} \cdots \partial_{a_0}$ (composing a la Eilenberg). The graded set of ends of A is denoted by EA). The domain of A is by definition, A-E(A). Similarly for the codomain of A (odd integers). Well formed simplicial sets are defined inductively: A zero dimensional well formed simplicial set is a singleton; an n-dimensional simplicial set is well formed if it's compatible and its domain and codomain are well formed simplicial sets. The collection of well formed simplicial sets form an $\omega$-catecgory (with union as composition).