Linear continuations and duality

Paul-André Melliès (1) , Nicolas Tabareau (1)

1 PPS - Preuves, Programmes et Systèmes

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.

Keywords: Linear logic; monoidal categories; tensorial negation; control categories; linear continuations

Domains: Category Theory [math.CT]; Programming Languages [cs.PL]

Paul-André Melliès, Nicolas Tabareau. Linear continuations and duality. 2007. ⟨hal-00339156⟩