one-dimensional simplicial submanifolds, which can be interpreted as hypermanifolds of constant time $t=0,1,2, ...$, endowing each triangulation with a distinguished notion of (proper) time. This time can be extended continuously to the interior of all triangles [@dl].

The ensemble of spacetimes forming the carrier space of the CDT path integral are all triangulations which consist of a fixed number $t_{tot}$ of triangulated strips $\Delta t=1$, where each strip is an arbitrary sequence of up- and down-triangles between times $t$ and $t+1$. The topology of space (usually chosen to be $S^1$) is not allowed to change in time, that is, branchings into multiple $S^1$-universes are forbidden. It was shown in [@benedettihensonmatrix] that the global foliation of a 1+1 dimensional CDT spacetime into such strips can be understood as consequence of a [*local*]{} regularity condition, namely, that precisely two spacelike edges be incident on any vertex. Note that these geometries are causally well behaved and obey a piecewise linear analogue of global hyperbolicity.

As already described in the introduction, the idea of creating individual path integral configurations with a well-behaved causal structure by imposing a preferred foliation on the underlying simplicial manifold is also realized in CDT in higher dimensions. Fig. \[fig:foliation\] 