#### Abstract

Finite cubical complexes are abstract models for parallel processing systems. The vertices of a complex K are the states of the system, and the execution paths are morphisms of the corresponding path category P(K).

The theory of path categories and path 2-categories for finite oriented cubical and simplicial complexes will be reviewed. There is an algorithm for computing the path category P(K) of a finite complex K which is based on its path 2-category. This 2-category algorithm will be displayed, and complexity reduction methods for the algorithm will be discussed.

The 2-category algorithm works well only for toy examples. The size of the path category P(K) of a complex K can be an exponential function of the size of K. The algorithm has so far resisted parallelization.

One wants combinatorial local to global methods for addressing examples that are effectively infinite.