  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.
