Let p: E \to X be a continuous map between locally compact Hausdorff spaces. Then CO_0(E) acquires the structure of a CO_0(X) module in an obvious way, given by (f \cdot u)(e) = f(p(e))u(e). Moreover, \rho (u)(x) = \sup _{e \in p^{-1}(x)}u(x) equips CO_0(E) with a Frechet module structure.

The bounded homomorphisms CO_0(E) \to CO_0(X) are in bijection with families of signed, real-valued Radon measures \mu _x \in CO_0(E_x)^* which vary continuously, in the sense that the resulting map X \to CO_0(E)^* (given by pushing forward along the inclusion E_x \hookrightarrow E) is weak-* continuous, and which are uniformly bounded in

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.

Let A = C_0(X) be a commutative C^*-algebra. A weakly smooth structure on A consists of a sub-*-algebra S \subseteq A satisfying the following properties:

1. S is dense in the norm topology of A.
2. If f: \mathbb {R}^n \to \mathbb {R} is a smooth function, and a_1, \dots a_n \in S, then f(a_1,\dots a_n) \in S, (in the sense of the continuous functions calculus).
3. S is equipped with a Frechet topology.

Given two weakly smooth structure (A,S), (B,S'), a C^*-homomorphism A \to B is said to preserve the smooth structure if it carries elements of S to S' and the restricted map S \to S' is continuous for the given topologies.

Let A be a real Banach algebra. Then a (real) Frechet module over A is a real vector space V equipped with a family of mappings \rho _i: V \to A^+ (where positivity is defined in the spectral sense), satisfying:

1. \rho _i(\alpha v) = \alpha \rho _i(v)
2. \rho _i(v + w) \leq \rho _i(v) + \rho _i(w). Again this is understood in the sense that \rho _i(v) + \rho _i(w)-\rho _i(v + w) has nonnegative spectrum.
3. If \rho _i(v)=0 for every i, then v = 0.

Given two Frechet modules, a module homomorphism f: V \to W is said to be bounded if, for every i, there exist K, \epsilon so that \rho _i^W(f(v)) \leq 1 if \rho _1^V(v), \dots ,\rho ^V_K(v) \leq \epsilon .

Note that in this definition we, a priori, allow \epsilon to be a generic element of A. Viewing A = C_\mathbb {R}(X) for some space X, this means the maps V_x \to W_x are each required to be bounded, but not uniformly bounded. But in fact since \epsilon must be strictly positive this requires it to be bounded away from 0, and so there is some uniform choice of \epsilon (say the infimum of the spectrum of the element \epsilon )

A tangent field---in a quite general sense---can be described as a particular type of operator C(X) \to C(X). The Hille-Yosida theorem, and others, give conditions for such an operator to give a transition semigroup T_t : t\in \mathbb {R}_{\geq 0}, which is a good notion of "integrable (stochastic) dynamical system". We are interested in bringing this perspective to categorical systems theory.

The first step is noting that by currying, such an operator can be viewed as a function X \to C(X)^*. Continuity (in the sense of landing in continuous functions) amounts to pointwise continuity (or weak-* continuity), although note that we will probably want to develop this theory for spaces X without a topology. Note also that we must abuse this a bit, because the operators are in general unbounded (and in particular partial), so the space C(X)^* must be replaced with some set of partial linear operators---which won't even be a vector space.

We can imagine that the structure on the space X induces some dense set of "sufficiently smooth" functions on which all these operators will be defined.

Let R be a commutative ring, A be an R-algebra, and let a \in A be an element. The spectrum of a, \sigma (a) \subseteq R, is the set of elements k so that a - k is not invertible.

Let (X, \Sigma ) be a measurable space. A \sigma -ideal is a subset N \subset \Sigma which is downwards stable (A \subset B, B \in N implies A \in N) and stable under countable unions.

A measurable space equipped with a \sigma -ideal (X,\Sigma ,N) is called an enhanced measurable space. Note that if (X,\Sigma ,\mu ) is a measure space, the collection of nullsets turns X into an enhanced measurable space. If there is no chance of confusion, we will use this implicitly.

We also use the terminology from measure spaces for enchanced measurable spaces---thus we refer to the sets in N as nullsets and say that two functions which differ only on a nullset are equal almost everywhere, and so on.