Giulio Lo Monaco | Searcharxiv

arXiv2021

Vopěnka's principle in $\infty$-categories

In this article, the interplay between Vopěnka's principle, as well as its weaker counterpart, and presentable $\infty$-categories is studied. Analogous statements, arising after replacing categories with $\infty$-categories in the original ones, are introduced and compared to these. Further, the attention is focused on the question of to what extent the consequences that (weak) Vopěnka's principle have on the detection of reflexive subcategories of a presentable categories can be generalized to $\infty$-categories.

math.CT↗

arXiv2019

Dependent products and 1-inaccessible universes

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.

math.CT↗