Lifting Star-Autonomy

For a functor \(\textsf{Q}\) from a category \(\textsf{C}\) to the category \(\textsf{Pos} \) of ordered sets and order-preserving functions, we study liftings—from the base \(\textsf{C}\) to the total category \(\int \textsf{Q}\) —of symmetric monoidal, symmetric monoidal closed, \(*\) -autonomous structures. Our systematic study relies on a bijection between liftings of functors to the total categories and some kind of lax-natural transformations, and yields exact conditions for these liftings. When \(\textsf{Q}\) factors as a monoidal functor through \(\textsf{SLatt} \) , the category of complete lattices and sup-preserving functions, we obtain, as a corollary of these conditions, that \(\int \textsf{Q}\) is closed. For such a \(\textsf{Q}\) , we also describe a method, analogous to the double negation nucleus from quantale theory, making it possible to quotient \(\textsf{Q}\) into a functor such that is \(*\) -autonomous. We prove then a representation theorem for those \(\textsf{Q}\) monoidal to \(\textsf{SLatt} \) for which \(\int \textsf{Q}\) is \(*\) -autonomous. The theory developed, originally motivated from the categories \(P\text {-}\textsf{Set} \) of Schalk and de Paiva, yields a generalization of Hyland and Schalk construction of \(*\) -autonomous categories by means of orthogonality structures.