Finitely presentable objects in \((Cb\text{-sets})_\mathrm{fs}\). (English) Zbl 07909550
Summary: 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 \(\operatorname{End}(\mathbb{D})\), 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.
Here, the interplay of two separate categories of finitely supported actions of a submonoid of \(\operatorname{End}(\mathbb{D})\), 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.
MSC:
| 08A30 | Subalgebras, congruence relations |
| 08C05 | Categories of algebras |
| 18A20 | Epimorphisms, monomorphisms, special classes of morphisms, null morphisms |
| 18C35 | Accessible and locally presentable categories |
| 20M30 | Representation of semigroups; actions of semigroups on sets |
| 68Q70 | Algebraic theory of languages and automata |
Keywords:
finitely supported \(M\)-sets; finitely supported \(Cb\)-sets; nominal sets; finitely presentable \(Cb\)-setsReferences:
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