We define and study notions of comprehension in (∞,1)-category theory. In essence, we do so by implementing Bénabou'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 (∞,1)-categories in terms of comprehension schemes.
Keywords: Comprehension, Relative Definability, Higher Category Theory, Fibered Category Theory, Univalent Mathematics
2020 MSC: 03G30, 18D30, 18N60, 18C50
Theory and Applications of Categories, Vol. 43, 2025, No. 6, pp 108-180.
Published 2025-03-09.
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