Linear continuations and duality

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.