Australian Category Seminar

An inductive geometric definition of k-dimensional source and target for an n-cube is described. Viewed as functions $n = \{1,2,...,n\} \to Λ = \{-,0,+\}$, the subcubes of the $n$-cube can be interpreted as words of length $n$ in $\{-,0,+\}$. The sources and targets of the n-cube arise as particular "k-blocks in $Λ^n$", where a "0-block" is an element of $Λ^n$, and we inductively define a k-block, k odd (resp. even) is a column (resp. row) of (k-1)-blocks. The sources $\theta_k(n)$, $\tau_k(n)$ of dimensions k of the n-cube can be defined inductively by the maps $\lambda_k, \nu_k, \mu_k : B_n^k \to B_{n + 1}^k, B_{n + 1}^k, B_{n + 1}^{k + 1}$ respectively, where $B_n^k$ is the set of k-blocks in Λ^n. Cocycle conditions for an n-category structure can be defined which correspond naturally to the cocycles discovered by Street in his work on orientals.