A Representation Theorem for Unique Decomposition Categories

Haghverdi introduced the notion of unique decomposition categories as a foundation for categorical study of Girard’s Geometry of Interaction (GoI). The execution formula in GoI provides a semantics of cut-elimination process, and we can capture the execution formula in every unique decomposition category: each hom-set of a unique decomposition category comes equipped with a partially defined countable summation, which captures the countable summation that appears in the execution formula. The fundamental property of unique decomposition categories is that if the execution formula in a unique decomposition category is always defined, then the unique decomposition category has a trace operator that is given by the execution formula. In this paper, we introduce a subclass of unique decomposition categories, which we call strong unique decomposition categories, and we prove the fundamental property for strong unique decomposition categories as a corollary of a representation theorem for strong unique decomposition categories: we show that for every strong unique decomposition category C, there is a faithful strong symmetric monoidal functor from C to a category with countable biproducts, and the countable biproducts characterize the structure of the strong unique decomposition category via the faithful functor.