Parametrized higher category theory

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