Lifting star-autonomous structures
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For a functor Q from a category C to the category Pos of ordered sets and order-preserving functions, we study liftings of various kind of structures from the base category C to the total(or Grothendieck) category ∫ Q. That lifting a monoidal structure corresponds to giving some lax natural transformation making Q almost monoidal, might be part of folklore in category theory.We rely on and generalize the tools supporting this correspondence so to provide exact conditions for lifting symmetric monoidal closed and star-autonomous structures.A corollary of these characterizations is that, if Q factors as a monoidal functor through SLatt, the category of complete lattices and sup-preserving functions, then ∫ Q is always symmetric monoidalclosed. In this case, we also provide a method, based on the double negation nucleus from quantale theory, to turn ∫ Q into a star-autonomous category.The theory developed, originally motivated from the categories P-Set of Schalk and de Paiva, yields a wide generalization of Hyland and Schalk construction of star-autonomous categories by means of orthogonality structures.
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