A category of FI type is one which is sufficiently similar to finite sets and injections 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.

A tremendous debt of gratitude is owed to Derek Lowenberg, who introduced me to representation stability and helped me carve out the ideas presented here. Reid Barton and Mike Shulman pointed out instructive examples which helped to identify necessary and sufficient conditions for the Grothendieck construction to produce an EI category. I would also like to thank John Baez, Jonathan Beardsley, Spencer Breiner, Scott Carter, Nir Gadish, Wee Liang Gan, David Jaz Myers, Todd Trimble, and Christina Vasilakopoulou for helpful discussions.