  We develop foundations for the category theory of $\infty$-categories
parametrized by a base $\infty$-category. Our main contribution is a theory of
indexed homotopy limits and colimits, which specializes to a theory of
$G$-colimits for $G$ a finite group when the base is chosen to be the orbit
category of $G$. We apply this theory to show that the $G$-$\infty$-category of
$G$-spaces is freely generated under $G$-colimits by the contractible
$G$-space, thereby affirming a conjecture of Mike Hill.
