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]).

Let \mathcal {A}: \mathcal {C}^\mathrm {op} \to \mathsf {Cat} be an indexed category. A chart is simply a morphism in the Grothendieck construction \int \mathcal {A}. A lens is a morphism in the fiberwise opposite \int \mathcal {A}(-)^\mathrm {op}.

Given an indexed category \mathcal {A}: \mathcal {C}^\mathrm {op} \to \mathsf {Cat}, the double category \mathsf {\mathbb Arena} of arenas has

1. Objects the objects of \int \mathcal {A}---note that these are the same as the objects of \int \mathcal {A}(-)^\mathrm {op}
2. Vertical morphisms the lenses
3. Horizontal morphisms the charts
4. The double category is thin. Given a square of this form

   

   we can first project it to a square in \mathcal {C}. If this commutes, we can pull the lenses and charts back to a square in \mathcal {A}(A_1). The above square of lenses and charts will be said to commute if both of these squares commute.

Let A be an arena. A dynamical system with interface A is an object S \in \mathcal {C} and a lens TS \leftrightarrows A. A morphism of systems is a map S \to S' so that this square commutes in \mathsf {\mathbb Arena}:

The category of systems with interface A is denoted \mathsf {Sys}(A)

Given a lens A \leftrightarrows A', postcomposition gives a functor \mathsf {Sys}(A) \to \mathsf {Sys}(A'). Given a chart A \rightrightarrows A', there is a profunctor \mathsf {Sys}(A) \nrightarrow \mathsf {Sys}(A'), where the set over TS \leftrightarrows A and TS' \leftrightarrows A' is the set of maps S \to S' so that this square commutes:

This defines a double functor \mathsf {Sys}: \mathsf {\mathbb Arena} \to \mathsf {\mathbb Cat}.

Let F: \mathsf {Set} \to \mathsf {Set} be a functor. Then there is a dynamical systems theory where \mathcal {C} = \mathsf {Set}, \mathcal {A} is constant at \mathsf {Set}, and T(X) = (X,F(X)). It is easy to see that the category of closed dynamical systems in this theory is equivalently the category of F-coalgebras.

There is a theory of dynamical systems with category of spaces \mathcal {C} = \mathsf {Set}, category of bundles \mathcal {A}(X) = \mathsf {Set}_{/X} (with pullbacks for reindexing), and tangent bundle TX = X \times X \xrightarrow {\pi _0} X. We call this the theory of discrete dynamical systems. (In the sense that they are both discrete-time and discrete-space).

In the theory of discrete dynamical systems:

1. The category of lenses \int \mathcal {A}(-)^\mathrm {op} is equivalently the category of polynomial functors and natural transformations between them.
2. Using this equivalence, the category of dynamical systems with interface p is exactly the category of p-coalgebras and homomorphisms

There is a systems theory where \mathcal {C} = \mathsf {Set}, \mathcal {A}(X) = Kl(\Delta )^X (with the evident reindexing maps) and T(X) = X \times X.

In this theory, a closed dynamical system is a set equipped with a map X \to \Delta (X). An open dynamical system has stochastic update function, but output which depends deterministically on the current system state.

There is a dynamical systems theory where \mathcal {C} = \mathsf {SmMfd} is the category of smooth manifolds, \mathcal {A}_X is the category of fiber bundles over X, with f^*: \mathcal {A}_Y \to \mathcal {A}_X given by pullback along f: X \to Y (note that pullbacks of bundles always exist), and with TX being the tangent bundle of X in the ordinary sense. In this case, closed dynamical systems are manifolds equipped with (smooth) vector fields, which is what is classically thought of as a smooth dynamical system.

Consider the theory of discrete-time, discrete dynamical systems from Example [efr-002F]. Let \xi : TS \leftrightarrows {I \choose O} be a system. Consider the system c: T\mathbb {N} \leftrightarrows {* \choose \mathbb {N}} given by n \mapsto n in the forwards direction, and (n,*) \mapsto n+1 in the backwards direction. (To be clear, the object denoted {* \choose \mathbb {N}} is the map \mathbb {N} \to \mathbb {N}---it is a singleton in each fiber). Then a morphism of systems c \to \xi consists of the following data:

1. A function x: \mathbb {N} \to S.
2. Another function o: \mathbb {N} \to O.
3. A third function i: \mathbb {N} \to I so that i(n) \in I_{o(n)}.
4. So that o(n) = \xi (x(n)) and \xi ^\#(x(n),i(n)) = x(n+1)

That is, it is a choice of a sequence x_n of points in the state-space, and a sequence of inputs i_n compatible with the outputs, so that this sequence obeys the dynamics. In other words, it is a trajectory of the system.