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(∞,1)-Categorical comprehension schemes
Theory and applications of categories, Lawvere Festschrift, Tome 43 (2025), pp. 108-180 Cet article a éte moissonné depuis la source Theory and Applications of Categories website

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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.
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Classification : 03G30, 18D30, 18N60, 18C50
Keywords: Comprehension, Relative Definability, Higher Category Theory, Fibered Category Theory, Univalent Mathematics 
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Raffael Stenzel. (∞,1)-Categorical comprehension schemes. Theory and applications of categories, Lawvere Festschrift, Tome 43 (2025), pp. 108-180. http://geodesic.mathdoc.fr/item/TAC_2025_43_a5/