$(\infty,1)$-Categorical Comprehension Schemes

Abstract

We define and study notions of comprehension in $(\infty,1)$-category theory. In essence, we do so by implementing B\'{e}nabou's foundations of naive category theory in a univalent meta-theory. In particular, we develop natural generalizations of smallness and relative definability in this context, and show for instance that the universal cartesian fibration is small. Furthermore, by building on Johnstone's notion of comprehension schemes for ordinary fibered categories, we characterize and relate numerous higher categorical properties and structures such as left exactness, local cartesian closedness, univalent morphisms and internal $(\infty,1)$-categories in terms of comprehension schemes.