# Identity crises between supercompactness and Vopenka's Principle

## TL;DR

This paper investigates the hierarchy of supercompactness notions related to Vopenka's Principle, demonstrating that the least supercompact can be strictly below or equal to certain $C^{(n)}$-supercompact cardinals, revealing complex identity phenomena.

## Contribution

It establishes the consistency of various identity crises between supercompactness and $C^{(n)}$-supercompactness, answering open questions in the field.

## Key findings

- The least supercompact can be strictly below the least $C^{(1)}$-supercompact.
- The least supercompact can also be $C^{(1)}$-supercompact.
- Ultimate identity crises between these notions are consistent under certain hypotheses.

## Abstract

In this paper we study the notion of $C^{(n)}$-supercompactness introduced by Bagaria in \cite{Bag} and prove the identity crises phenomenon for such class. Specifically, we show that consistently the least supercompact is strictly below the least $C^{(1)}$-supercompact but also that the least supercompact is $C^{(1)}$-supercompact (and even $C^{(n)}$-supercompact). Furthermore, we prove under suitable hypothesis that the ultimate identity crises is also possible. These results solve several questions posed by Bagaria and Tsaprounis.