Tautological systems theory [efr-2ZUR]

Let \mathcal {E} \to \mathcal {A} be a fibration. Then also \mathcal {E} \times _\mathcal {A} \mathcal {E} \to \mathcal {E} is a fibration. Moreover it admits a canonical section given by the diagonal. If \mathcal {E} \to \mathcal {A} is a monoidal fibration, this structure passes to this pullback (and the diagonal becomes monoidal).

In the context of Categorical Systems Theory, this gives a "tautological systems theory", where each space is preequipped with a "tangent bundle" (and each map is equipped with extra structure preserving this). Note given a section T: \mathcal {A} \to \mathcal {E}, there is induced \mathcal {E} \to \mathcal {E} \times _\mathcal {A} \mathcal {E} which becomes a morphism of systems theories.