Musings around the geometry of interaction, and coherence

We introduce the Danos-Régnier category DR(M) of a linear inverse monoid M, as a categorical description of geometries of interaction (GOI) inspired from the weight algebra. The natural setting for GOI is that of a so-called weakly Cantorian linear inverse monoid, in which case DR(M) is a kind of symmetrized version of the classical Abramsky-Haghverdi-Scott construction of a weak linear category from a GOI situation. It is well-known that GOI is perfectly suited to describe the multiplicative fragment of linear logic, and indeed DR(M) will be a ∗-autonomous category in this case. It is also well-known that the categorical interpretation of the other linear connectives conflicts with GOI interpretations. We make this precise, and show that DR(M) has no terminal object, no Cartesian product of any two objects, and no exponential-whatever M is, unless M is trivial. However, a form of coherence completion of DR(M) à la Hu-Joyal (which for additives resembles a layered approach à la Hughes-van Glabbeek), provides a model of full classical linear logic, as soon as M is weakly Cantorian. One finally notes that Girard's notion of coherence is pervasive, and instrumental in every aspect of this work.