  In 1892, Lyapunov provided a fundamental contribution to stability theory by
introducing so-called Lyapunov functions and Lyapunov equilibria. He
subsequently showed that, for linear systems, the two concepts are equivalent.
These concepts have since been extended to diverse types of dynamical systems,
and in all settings the equivalence remains valid. However, this involves an
often technical proof in each new setting where the concepts are introduced. In
this article, we investigate a categorical framework where these results can be
unified, exposing a single underlying reason for the equivalence to hold in all
cases. First we define what is a dynamical system. Then we introduce the notion
of a level-set morphism, which in turn allows us to define the concepts of a
Lyapunov equilibrium and a Lyapunov function in a categorical setting. We
conclude by a proof of their equivalence.
