Relative $ω$-stability, relative categoricity and internal covers

We study a special case of a relatively categorical theory $(T,P)$, namely when $T$ is an internal cover of $T^{P}$. We give a structure theory for relatively categorical internal covers. After passing to $T^{eq}$ and naming a parameter from the $P$-part, they are precisely the "pure torsor covers" of $T^{P}$, obtained simply by adjoining a new sort $S$ for a principal homogeneous space or torsor of an $\emptyset$-definable group $G$ in $T^{P}$, and a symbol for the action. We prove that $T$ is relatively $\omega$-stable (or $\omega$-stable over $P$) as defined in [4] iff the dual binding group $H$ has the $DCC$ on definable subgroups.