Quantitative Semantics Revisited (extended Abstract)

In the coherence space semantics of linear logic, the webs of the spaces interpreting the exponentials may be deened using multi-cliques (multisets whose supports are cliques) instead of cliques. Inspired by the quantitative semantics of Jean-Yves Girard, we give a characterization of the morphisms of the co-Kleisly category of the corresponding comonad (this category is cartesian closed and, therefore, is a model of intuitionistic logic). It turns out that these morphisms are the convex and multiplicative functions mapping multicliques to multicliques. This characterization is achieved via a normal form theorem, which associates a trace to each such map.