Semigroup generator operators as tangent fields [efr-86YE]

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.