Extensions of representation stable categories

A category of FI type is one which is sufficiently similar to finite sets and injections so as to admit nice representation stability results. Several common examples admit a Grothendieck fibration to finite sets and injections. We begin by carefully reviewing the theory of fibrations of categories with motivating examples relevant to algebra and representation theory. We classify which functors between FI type categories are fibrations, and thus obtain sufficient conditions for an FI type category to be the result of a Grothendieck construction.