Parametrized higher category theory

Abstract

We develop foundations for the category theory of [infinity]-categories parametrized by a base occategory. Our main contribution is a theory of parametrized homotopy limits and colimits, which recovers and extends the Dotto-Moi 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-[infinity]-category of G-spaces is freely generated under G-colimits by the contractible G-space, thereby affirming a conjecture of Mike Hill.