than the required two light cones at a vertex, as illustrated by the local neighbourhoods depicted in Fig. \[fig:causality2d\]. Local causality implies crossing exactly two light cones when going around a vertex or, equivalently, crossing exactly four lightlike lines emanating radially from the vertex.

As a next step, we will consider the global causal structure of individual triangulated manifolds satisfying the local vertex causality condition everywhere. This global structure will in general depend on the chosen topology. For simplicity, we will restrict ourselves to the cases $[0,1]\times S^1$ (space is a compact circle) and $[0,1]\times [0,1]$ (space is a closed interval), where the initial and final boundary are assumed to be spacelike, and any other boundaries (in the second case) timelike. We will call such a spacetime globally causal if it does not contain any closed timelike curves. A “timelike curve" for our purposes will be a sequence of oriented, timelike links in a triangulation. Since the interior of every Minkowskian triangle has a well-defined light cone structure, a choice of orientation (i.e. a choice of which one is the past and which one the future light cone) induces an orientation on its timelike edges, which can be captured by 