Uniqueness of actions in algebraically coherent categories

In the category of groups, an action of $C$ on $X$ is uniquely determined by a morphism $C\to \mathrm{Aut}(X)$. In particular, if $A,B$ are subobjects of $C$ such that $A\vee B=C$, then an action is determined by its restrictions to $A$ and $B$.

The notion of internal action can be generalized to any semi-abelian category [1]; but in general this fact is no longer true. In [4], the authors introduced the uniqueness of actions condition for semi-abelian categories :

• $(\mathrm{UA})$

  given a jointly strongly epimorphic cospan , then for any actions ${\xi }_1,{\xi }_2,{\xi }_3,{\xi }_4$ on $X$, if the diagram

  commutes, then ${\xi }_3={\xi }_4$.

Thus the category of groups, and more generally any category of interest, satisfies $(\mathrm{UA})$.

We show that if a semi-abelian category $𝒞$ is algebraically coherent [3], then it satisfies a restricted version of $(\mathrm{UA})$; namely, the condition above holds for jointly epimorphic cospans such that one leg has normal image. We also show that this restricted case is already sufficient to obtain several applications of the $(\mathrm{UA})$ condition previously considered, such as those of [4, 2].