Definition Theory of Dynamical Systems [efr-000A]

A theory of dynamical systems is an indexed category \mathcal {A}: \mathcal {C}^\mathrm {op} \to \mathsf {Cat} equipped with a section of its Grothendieck construction T: \mathcal {C} \to \int \mathcal {A}.

It is called monoidal if \mathcal {C} is a monoidal category, and \mathcal {A} is a lax monoidal functor. It is further called symmetric monoidal if \mathcal {C} is a symmetric monoidal category and \mathcal {A} is a symmetric lax monoidal functor. Note that this is equivalent to requiring \int \mathcal {A} \to \mathcal {C} to be a (symmetric) monoidal fibration in the sense of Reference [shulman-monfibs] (see also Reference [moeller-vasilakopoulou]).