The purpose of this writing is to explore the exact relationship running between geometric ∞-toposes and Mike Shulman's proposal for the notion of elementary ∞-topos, and in particular we will focus on the set-theoretical strength of Shulman's axioms, especially on the last one dealing with dependent sums and products, in the context of geometric ∞-toposes. Heuristically, we can think of a collection of morphisms which has a classifier and is closed under these operations as a well-behaved internal universe in the ∞-category under consideration. We will show that this intuition can in fact be made to a mathematically precise statement, by proving that, once fixed a Grothendieck universe, the existence of such internal universes in geometric ∞-toposes is equivalent to the existence of smaller Grothendieck universes inside the bigger one. Moreover, a perfectly analogous result can be shown if instead of geometric ∞-toposes our analysis relies on ordinary sheaf toposes, although with a slight change due to the impossibility of having true classifiers in the 1-dimensional setting. In conclusion, it will be shown that, under stronger assumptions positing the existence of intermediate-size Grothendieck universes, examples of elementary ∞-toposes with strong universes which are not geometric can be found.
Keywords: higher categories, higher toposes, elementary higher toposes, Grothendieck universes, large cardinals, dependent products, dependent sums, classifiers, generic morphisms
2020 MSC: Primary: 03E55, 18N60
Theory and Applications of Categories, Vol. 37, 2021, No. 5, pp 107-143.