  Let $G$ and $A$ be objects of a finitely cocomplete homological category
$\mathbb C$. We define a notion of an (internal) action of $G$ of $A$ which is
functorially equivalent with a point in $\mathbb C$ over $G$, i.e. a split
extension in $\mathbb C$ with kernel $A$ and cokernel $G$. This notion and its
study are based on a preliminary investigation of cross-effects of functors in
a general categorical context. These also allow us to define higher categorical
commutators. We show that any proper subobject of an object $E$ (i.e., a kernel
of some map on $E$ in $\mathbb C$) admits a "conjugation" action of $E$,
generalizing the conjugation action of $E$ on itself defined by Bourn and
Janelidze. If $\mathbb C$ is semi-abelian, we show that for subobjects $X$, $Y$
of some object $A$, $X$ is proper in the supremum of $X$ and $Y$ if and only if
$X$ is stable under the restriction to $Y$ of the conjugation action of $A$ on
itself. This amounts to an elementary proof of Bourn and Janelidze's functorial
equivalence between points over $G$ in $\mathbb C$ and algebras over a certain
monad $\mathbb T_G$ on $\mathbb C$. The two axioms of such an algebra can be
replaced by three others, in terms of cross-effects, two of which generalize
the usual properties of an action of one group on another.
