Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585321120240701Finitely presentable objects in ${\rm(}Cb\text{-}{\bf Sets}{\rm)}_{_{\rm fs}}$17520910461510.48308/cgasa.2024.235466.1487ENMahdieh HaddadiDepartment of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran.Khadijeh KeshvardoostDepartment of Mathematics, Velayat University, Iranshahr, Sistan and
Balochistan, Iran.Aliyeh HosseinabadiFaculty of Mathematics, Statistics and Computer Sciences, Department of Mathematics, Semnan University, Semnan, Iran.Journal Article20240425Pitts 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. <br />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.<br />Then, in the category of finitely supported $Cb$-sets, we characterize the finitely presentable objects and provide a generator in this category.https://cgasa.sbu.ac.ir/article_104615_923d1567db715dc3f19bcc09398d001c.pdf