  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 natural 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, logical choice
via tests in a Boolean algebra, and input from a stream of values. Finally, we
exploit our framework to describe a normalisation-by-trace-evaluation process
for behaviours of various kinds of coalgebraic generative systems, including
labelled transition systems, probabilistic generative systems, and stream
processors.
