Lifting star-autonomous structures

TL;DR

This paper characterizes how to lift monoidal and star-autonomous structures from a base category to a total category via functors into Pos, generalizing existing constructions and providing conditions for such liftings.

Contribution

It offers exact conditions for lifting symmetric monoidal closed and star-autonomous structures, extending previous work and introducing a method using double negation nuclei for star-autonomous categories.

Findings

If Q factors through SLatt, the total category is symmetric monoidal closed.

Provides a method to turn the total category into a star-autonomous category.

Generalizes Hyland and Schalk's construction using orthogonality structures.

Abstract

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 $\int Q$. 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 $\int Q$ is always symmetric monoidalclosed. In this case, we also provide a method, based on the double negation nucleus from quantale theory, to turn $\int Q$ 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.