Nonpairwise "higher-order" network interactions can shape the collective
network dynamics of interacting nodes. But what is the right
"higher-order" object to encode these interactions? For examples of
synchronization and heteroclinic dynamics, we start with the dynamical
phenomenon and give conditions what type of higher-order interactions
are required for the dynamics to arise. This highlights limitations of
using classical hypergraphs to model higher-order interactions.