  A central theme in current work in quantum information and quantum
foundations is to see quantum mechanics as occupying one point in a space of
possible theories, and to use this perspective to understand the special
features and properties which single it out, and the possibilities for
alternative theories. Two formalisms which have been used in this context are
operational theories, and categorical quantum mechanics. The aim of the present
paper is to establish strong connections between these two formalisms. We show
how models of categorical quantum mechanics have representations as operational
theories. We then show how nonlocality can be formulated at this level of
generality, and study a number of examples from this point of view, including
Hilbert spaces, sets and relations, and stochastic maps. The local, quantum,
and no-signalling models are characterized in these terms.
