Weak action representability of 2-nilpotent groups

Abstract

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 ${\mathrm{Aut}}_c(X)$ of central automorphisms of $X$. We also characterise the morphisms $B\to {\mathrm{Aut}}_c(X)$ 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 ${\mathrm{Aut}}_c(X)$, 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.