  Random attractors are the time-evolving pullback attractors of stochastically
perturbed, deterministically chaotic dynamical systems. These attractors have a
structure that changes in time, and that has been characterized recently using
{\sc BraMAH} cell complexes and their homology groups. This description has
been further improved for their deterministic counterparts by endowing the cell
complex with a directed graph, which encodes the order in which the cells in
the complex are visited by the flow in phase space. A templex is a mathematical
object formed by a complex and a digraph; it provides a finer description of
deterministically chaotic attractors and permits their accurate classification.
In a deterministic framework, the digraph of the templex connects cells within
a single complex for all time. Here, we introduce the stochastic version of a
templex. In a random templex, there is one complex per snapshot of the random
attractor and the digraph connects the generators or ``holes'' of successive
cell complexes. Tipping points appear in a random templex as drastic changes of
its holes in motion, namely their birth, splitting, merging, or death. This
paper introduces and computes the random templex for the noise-driven Lorenz
system's random attractor (LORA).
