Monoidal Indeterminates and Categories of Possible Worlds

Given any symmetric monoidal category C, a small symmetric monoidal category Σ and a strong monoidal functor j:Σ→C, it is shown how to construct C[x:jΣ], a polynomial such category, the result of freely adjoining to C a system x of monoidal indeterminates for every object j(w) with w∈Σ satisfying a naturality constraint with the arrows of Σ. As a special case, we show how to construct the free co-affine category (symmetric monoidal category with initial unit) on a given small symmetric monoidal category. It is then shown that all the known categories of “possible worlds” used to treat languages that allow for dynamic creation of “new” variables, locations, or names are in fact instances of this construction and hence have appropriate universality properties.