Definition [lcc-003H]

Let \mathcal {A}: \mathcal {C}^\mathrm {op} \to \mathsf {Cat}, T: \mathcal {C} \to \int \mathcal {A} be a theory of dynamical systems in the sense of Categorical Systems Theory. Let A,B \in \int \mathcal {A} be arenas. Then a controlled cybernetic system A \nrightarrow B is a tuple consisting of a third interface X, a system TS \to X, and a lens A \otimes X \leftrightarrow Y

It may seem that we should ask simply for a morphism A \otimes TS \to B. However, the above definition turns out to be more natural. Since there are multiple interacting notions of reparametrization for a controlled cybernetic system, it is not at all obvious that we can always feel free to replace every system with one of this simplified form (as we would be able to for parametrized morphisms), hence we choose this definition.