Complete Axioms for Categorical Fixed-Point Operators
LICS '00 Proceedings of the 15th Annual IEEE Symposium on Logic in Computer Science
Geometry of Interaction and linear combinatory algebras
Mathematical Structures in Computer Science
A categorical model for the geometry of interaction
Theoretical Computer Science - Automata, languages and programming: Logic and semantics (ICALP-B 2004)
Towards a typed geometry of interaction
Mathematical Structures in Computer Science
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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.