  This paper investigates, in the context of discrete-time switching systems,
the problem of comparison for path-complete stability certificates. We
introduce and study abstract operations on path-complete graphs, called lifts,
which allow us to recover previous results in a general framework. Moreover,
this approach highlights the existing relations between the analytical
properties of the chosen set of candidate Lyapunov functions (the template) and
the admissibility of certain lifts. This provides a new methodology for the
characterization of the order relation of path-complete Lyapunov functions
criteria, when a particular template is chosen. We apply our results to
specific templates, notably the sets of primal and dual copositive norms,
providing new stability certificates for positive switching systems. These
tools are finally illustrated with the aim of numerical examples.
