Finitely presentable objects in (Cb-Sets) fs

Pitts generalized nominal sets to finitely supported Cb-sets by utilizing the monoid Cb of name substitutions instead of the monoid of finitary permutations over names. Finitely supported Cb-sets provide a framework for studying essential ideas of models of homotopy type theory at the level of convenient abstract categories. Here, the interplay of two separate categories of finitely supported actions of a submonoid of End(D), for some countably infinite set D, over sets is first investigated. In particular, we specify the structure of free objects. Then, in the category of finitely supported Cb-sets, we characterize the finitely presentable objects and provide a generator in this category.