Finitely presentable objects in {\rm(}Cb\text{-}{\bf Sets}{\rm)}_{_{\rm 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 {\rm End}(\mathbb {D}), for some countably infinite set \mathbb {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.

Keywords:

Finitely supported M-sets , Finitely supported Cb-sets , nominal sets , finitely presentable Cb-sets