Let $G$ be the group acting on two sets $X, Y$. Let $G_x$ and $G_y$ be stabilizer subgroups of some elements $x \in X, y \in Y$. It is easy to see that $G_x \cap G_y = G_{(x,y)}$, when we combine two actions in a standard way to get a $G$-action on $X \times Y$.
I'm interested if there is a similar "natural" interpretation for the join of $G_x$ and $G_y$?
Let $f\colon G\rightarrow X$ be the map $f(g)=gx$, and $h\colon G\rightarrow Y$ the map $h(g)=gy$. Let $Z=X\coprod_{x=y} Y$ be the push-out or colimit of the diagram made out of $f$ and $h$ in the category of $G$-sets. Let $i\colon X\rightarrow Z$ and $j\colon Y\rightarrow Z$ be the maps that come along with $Z$. Then $i(x)=j(y)$, and the stabilizer of this point $z=i(x)=j(y)$ of $Z$ is the join of $G_x$ and $G_y$ in $G$.