Injectivity of \(G\)-nominal sets. (English) Zbl 1401.20006
Summary: In this paper, we consider injectivity for nominal sets, \(G\)-sets for a subgroup \(G\) of the group \(\operatorname{Perm}(\mathbb{D})\) of permutations on an infinite countable set \(\mathbb{D}\), with
finitely supported elements. Also, we study injectivity of nominal sets with respect to monomorphisms whose domains or codomains are single-orbit. Furthermore, we define the category of \(G_C\)-sets, for finite subsets \(C\) of \(\mathbb{D}\), where \(G_C\) is the set of elements of \(G\) fixing the elements of \(C\), and study injectivity in such categories.
MSC:
20B27 | Infinite automorphism groups |
08B30 | Injectives, projectives |
18A20 | Epimorphisms, monomorphisms, special classes of morphisms, null morphisms |
18G05 | Projectives and injectives (category-theoretic aspects) |