Monoidal Indeterminates

Abstract

Given any symmetric monoidal
category $\C$ and a sub-smc $\Sigma$,
it is shown how to construct~$\C[\Sigma]$,
a \emph{polynomial} such category, the result of freely adjoining
to $\C$ a system of monoidal indeterminates for every object of $\Sigma$
satisfying a naturality constraint with the arrows of
$\Sigma$. As a special case, we show how to construct the free coaffine
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.