consecutive “slabs", as described earlier. The presence of this time label allows us to define a transfer matrix and an associated propagator with the correct behaviour under composition (whose continuum limit in dimension 1+1 can be computed analytically [@al]), and prove a version of reflection positivity [@3d4d]. Yet another advantage of having an explicit time variable is that we can construct observables like the volume profile, which measures the distribution of spatial volume as a function of time. The analysis of these volume profiles in 3+1 dimensions has been crucial in relating the large-scale behaviour of CDT to a de Sitter cosmology in the continuum [@desitter; @semiclassical].[^3]

By contrast, in the enlarged CDT set-up we will be considering, typical triangulations will be more complicated, in the sense that the purely spatial subsimplices of codimension 1 will no longer align themselves into a neat sequence of simplicial hyper[*manifolds*]{}, but instead will form branching structures, as will be explained in more detail below. This also implies that we no longer have a distinguished time variable at our disposal. Nevertheless, as we shall demonstrate for the nontrivial case of 2+1 dimensions, it is possible to construct a meaningful time variable, whose restriction to 