  An open stochastic system \`a la Willems is a system affected two
qualitatively different kinds of uncertainty -- one is probabilistic
fluctuation, and the other one is nondeterminism caused by lack of information.
We give a formalization of open stochastic systems in the language of category
theory. A new construction, which we term copartiality, is needed to model the
propagating lack of information (which corresponds to varying sigma-algebras).
  As a concrete example, we discuss extended Gaussian distributions, which
combine Gaussian probability with nondeterminism and correspond precisely to
Willems' notion of Gaussian linear systems. We describe them both as
measure-theoretic and abstract categorical entities, which enables us to
rigorously describe a variety of phenomena like noisy physical laws and
uninformative priors in Bayesian statistics. The category of extended Gaussian
maps can be seen as a mutual generalization of Gaussian probability and linear
relations, which connects the literature on categorical probability with ideas
from control theory like signal-flow diagrams.
