Hypernormalisation in an abstract setting

Jacobs' hypernormalisation is a construction on finitely supported discrete probability distributions, obtained by generalising certain patterns occurring in quantitative information theory. In this paper, we generalise Jacobs' notion in turn, by describing a notion of hypernormalisation in the abstract setting of a symmetric monoidal category endowed with a linear exponential monad—a structure arising in the categorical semantics of linear logic. We show that Jacobs' hypernormalisation arises in this fashion from the finitely supported probability measure monad on the category of sets, which can be seen as a linear exponential monad with respect to a non-standard monoidal structure on sets which we term the convex monoidal structure. We give the construction of this monoidal structure in terms of a quantum-algebraic notion known as a tricocycloid. Besides the motivating example, and its generalisations to the continuous context, we give a range of other instances of our abstract hypernormalisation, which swap out the side-effect of probabilistic choice for other important side-effects such as non-deterministic choice, ranked choice, and input from a stream of values.