Free functor from the category of G-nominal sets to that of 01-G-nominal sets

Abstract

The category $\mathbf{Nom}$, of all finitely supported G-sets, called G-nominal sets, where $G=\mathrm{Perm}_{\mathrm{f}}({\mathbb {D}})$ is the group of all finitary permutations over a countable infinite set ${\mathbb {D}}$, is a subject of interest by both set theorists and computer scientists. The category 01-Nom, of all G-nominal sets equipped with the source and the target operations, was introduced by Pitts. He has shown that this category is isomorphic to the category of sets whose elements have a finite support property with respect to an action of the monoid Cb of name substitutions. The latter category is a coreflective subcategory of the category $\mathbf{Set}^{Cb}$, of sets with the action of the monoid Cb. For a functorial relation between the categories $\mathbf{Nom}$ and $\mathbf{Set}^{Cb}$, we study the existence of the free objects in the category 01-Nom. More precisely, we construct the left adjoint to the forgetful functor from the category of 01-G-nominal sets to the category of G-nominal sets, where G is a suitable subgroup of the group of all permutations over a countable infinite set ${\mathbb {D}}$.