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In 1892, Lyapunov provided a fundamental contribution to stability theory byintroducing so-called Lyapunov functions and Lyapunov equilibria. Hesubsequently 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 anoften technical proof in each new setting where the concepts are introduced. Inthis article, we investigate a categorical framework where these results can beunified, exposing a single underlying reason for the equivalence to hold in allcases. First we define what is a dynamical system. Then we introduce the notionof a level-set morphism, which in turn allows us to define the concepts of aLyapunov equilibrium and a Lyapunov function in a categorical setting. Weconclude by a proof of their equivalence.

A categorical view on the converse Lyapunov theorem