Vop\v{e}nka's principle in $\infty$-categories

TL;DR

This paper explores how Vopěnka's principle and its weaker form relate to presentable ∞-categories, examining their implications for detecting reflexive subcategories and introducing analogous statements in the ∞-categorical context.

Contribution

It introduces and compares ∞-categorical analogs of Vopěnka's principle and investigates their impact on reflexive subcategories in presentable ∞-categories.

Findings

Analogous statements to Vopěnka's principle in ∞-categories are formulated.

The extent of Vopěnka's principle's consequences on reflexive subcategories is analyzed.

Comparisons between classical and ∞-categorical versions of the principles are provided.

Abstract

In this article, the interplay between Vop\v{e}nka'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) Vop\v{e}nka's principle have on the detection of reflexive subcategories of a presentable categories can be generalized to $\infty$-categories.