Ideals of closed categories

AbstractA complete closed poset can be viewed as a commutative monoid in the closed category SI of complete lattices and sup-preserving maps. A lax adjunction between closed posets and the 2-category CSI of symmetric, monoidal closed categories over sup-lattices is described. This makes use of categories of ‘modules’ over a closed poset. If V is a suitably complete and cocomplete symmetric monoidal closed category, it is shown that the subobjects of the unit for ⊗ in V form a closed poset. The functoriality of this ideal construction is investigated; it is functorial in two different ways depending upon the type of morphism we consider for our closed categories. It alternately provides a right lax or right colax adjoint to the inclusion of closed posets into the 2-category of closed categories under consideration