Weak action representability of 2-nilpotent groups
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| Format: | Preprint |
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2026
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| author | Burgio, Alessandro Dioguardi Mancini, Manuel Van der Linden, Tim |
| author_facet | Burgio, Alessandro Dioguardi Mancini, Manuel Van der Linden, Tim |
| contents | In this article, we investigate the representability of actions of the category $\mathsf{Nil}_2(\mathsf{Grp})$ of $2$-nilpotent groups. We first provide an algebraic characterisation of derived actions in $\mathsf{Nil}_2(\mathsf{Grp})$ by determining a universal strict general actor of an object $X$, which turns out to be the group $\operatorname{Aut}_c(X)$ of central automorphisms of $X$. We also characterise the morphisms $B \to \operatorname{Aut}_c(X)$ that define an action of $B$ on $X$ in $\mathsf{Nil}_2(\mathsf{Grp})$.
We then show that $\mathsf{Nil}_2(\mathsf{Grp})$ is not action representable, and that the existence of a weak representation is related to the amalgamation property. Using the construction of an amalgam of a suitable family of abelian subgroups of $\operatorname{Aut}_c(X)$, we prove that the category $\mathsf{Nil}_2(\mathsf{Grp})$ is weakly action representable, and that a weak representing object can be chosen to be an abelian group.
Finally, we show that $\mathsf{Nil}_2(\mathsf{Grp})$ is not locally algebraically cartesian closed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_22578 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weak action representability of 2-nilpotent groups Burgio, Alessandro Dioguardi Mancini, Manuel Van der Linden, Tim Category Theory Group Theory 08A35, 08C05, 16W22, 17A36, 18E13, 20F18, 20F28 In this article, we investigate the representability of actions of the category $\mathsf{Nil}_2(\mathsf{Grp})$ of $2$-nilpotent groups. We first provide an algebraic characterisation of derived actions in $\mathsf{Nil}_2(\mathsf{Grp})$ by determining a universal strict general actor of an object $X$, which turns out to be the group $\operatorname{Aut}_c(X)$ of central automorphisms of $X$. We also characterise the morphisms $B \to \operatorname{Aut}_c(X)$ that define an action of $B$ on $X$ in $\mathsf{Nil}_2(\mathsf{Grp})$. We then show that $\mathsf{Nil}_2(\mathsf{Grp})$ is not action representable, and that the existence of a weak representation is related to the amalgamation property. Using the construction of an amalgam of a suitable family of abelian subgroups of $\operatorname{Aut}_c(X)$, we prove that the category $\mathsf{Nil}_2(\mathsf{Grp})$ is weakly action representable, and that a weak representing object can be chosen to be an abelian group. Finally, we show that $\mathsf{Nil}_2(\mathsf{Grp})$ is not locally algebraically cartesian closed. |
| title | Weak action representability of 2-nilpotent groups |
| topic | Category Theory Group Theory 08A35, 08C05, 16W22, 17A36, 18E13, 20F18, 20F28 |
| url | https://arxiv.org/abs/2604.22578 |