Linear continuations and duality

Abstract : One fundamental aspect of linear logic is that its conjunction behaves in the same way as a tensor product in linear algebra. Guided by this intuition, we investigate the algebraic status of disjunction -- the dual of conjunction -- in the presence of linear continuations. We start from the observation that every monoidal category equipped with a tensorial negation inherits a lax monoidal structure from its opposite category. This lax structure interprets disjunction, and induces a multicategory whose underlying category coincides with the kleisli category associated to the continuation monad. We study the structure of this multicategory, and establish a structure theorem adapting to linear continuations a result by Peter Selinger on control categories and cartesian continuations.
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2007
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Paul-André Melliès, Nicolas Tabareau. Linear continuations and duality. 2007. 〈hal-00339156〉

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