Dependent products and 1-inaccessible universes

TL;DR

This paper explores the set-theoretic strength of axioms for elementary ∞-toposes, showing that certain conditions imply the existence of 1-inaccessible universes, which are stronger than inaccessible cardinals.

Contribution

It establishes a connection between Shulman's axioms for ∞-toposes and large cardinal assumptions, specifically 1-inaccessible universes, revealing set-theoretic implications.

Findings

Every geometric ∞-topos satisfying Shulman's axioms implies a 1-inaccessible universe.

The existence of 1-inaccessible cardinals leads to examples of non-geometric Shulman ∞-toposes.

Analogous results hold for ordinary sheaf toposes.

Abstract

The purpose of this writing is to show that, if we use the definition of elementary $\infty$-topos that has been proposed by Mike Shulman, then the fact that every geometric $\infty$-topos satisfies the required axioms, more specifically the last one of them, is actually something close to a large cardinal assumption. Putting it precisely, we will show that, once a Grothendieck universe has been chosen, the fact that every geometric $\infty$-topos satisfies Shulman's axioms is equivalent to saying that the Grothendieck universe was 1-inaccessible to start with, a condition which is strictly stronger than just being inaccessible. Moreover, a perfectly analogous result can be shown if instead of geometric $\infty$-toposes our analysis relies on ordinary sheaf toposes. In conclusion, it will be shown that, under stronger assumptions positing the existence of 1-inaccessible cardinals inside the Grothendieck universe, examples of Shulman $\infty$-toposes which are not geometric can be found.