Higher dimensional automata (HDA) have been widely studied as models of concurrent processes. Most current work focuses on developing the directed algebraic topological notions required to analyse HDA to determine their computer scientific properties (deadlock, safety, unreachable states, etc). Instead, this paper is concerned with the software engineering of HDA and details the specification of HDA by process algebra operations. The specifications work for cubical HDA, but are designed to also work for the explicit choice higher dimensional automata (ECHDA) originally proposed by Buckland. We introduce ω-multigraphs, a graphical notion which is easier to use than pasting schemes but more general than cubical complexes, we describe basic process algebra operations on ECHDA as constructions on ω-multigraphs, we discuss the trichotomy of concurrent, conferring, and conflicting choices, and note that the “deadlock choice” can arise from intersecting conflicting choices, and we remark that common software engineering refinements correspond to choice refinement.