Separated finitely supported Cb-sets

The monoid Cb of name substitutions and the notion of finitely supported Cb-sets introduced by Pitts as a generalization of nominal sets. A simple finitely supported Cb-set is a one point extension of a cyclic nominal set. The support map of a simple finitely supported Cb-set is an injective map. Also, for every two distinct elements of a simple finitely supported Cb-set, there exists an element of the monoid Cb which separates them by making just one of them into an element with the empty support. In this paper, we generalize these properties of simple finitely supported Cb-sets by modifying slightly the notion of the support map; defining the notion of 2-equivariant support map; and introducing the notions of s-separated and z-separated finitely supported Cb-sets. We show that the notions of s-separated and z-separated coincide for a finitely supported Cb-set whose support map is 2-equivariant. Among other results, we find a characterization of simple s-separated (or z-separated) finitely supported Cb-sets. Finally, we show that some subcategories of finitely supported Cb-sets with injective equivariant maps which constructed applying the defined notions are reflective.