and after $z$ in $h$. If $z = w^{-1}$ then $z^{-1}$ is the curve preceding $w$ in the geodesic whose domain is ${\mathcal{C}}(S \backslash w^{-1})$. If $z = w^{+1}$ then $z^{+1}$ is the curve following $w$ in the geodesic whose domain is ${\mathcal{C}}(S \backslash w^{+1})$ (see Figure \[general hierarchy\] (top)).

We can get a path in ${\mathcal{P}}_{rel}(S)$ by a similar process as before - going along each of the edges. Whenever we come across a domain separating curve, $c$, where $c$ is in the main geodesic or in a geodesic whose domain is ${\mathcal{C}}(S \backslash w)$ where $w$ is in the main geodesic, we add in the point $p_c$ into the path before moving on. For an example see Figure \[general hierarchy\]. These paths are *relative 3-archies*. As before, we will blur the line between the collection of geodesics and the path of a relative 3-archy.

When discussing hierarchies (or relative 3-archies), subsurface projections of curves or geodesics are involved. The following maps are to define what is meant by subsurface projections [@MMII]. An *essential subsurface* is a subsurface where each boundary component is essential.

Let ${\mathscr{P}}(X)$ be the set of subsets of $X$. For a set $A$ we define 