Ideals of closed categories (Q1105688)

By an ideal of a (symmetric monoidal) closed category, the authors mean a subobject of the unit for the monoidal structure. The main result of this paper is that, for a class of ``suitable'' closed categories, the operation which sends a closed category to its poset of ideals can be made into a functor, which is right lax (or colax) adjoint to the inclusion functor from closed posets to closed categories. This generalizes known results for categories of modules over commutative rings and for Grothendieck toposes.