# Inner horns for 2-quasi-categories

## TL;DR

This paper establishes a more manageable combinatorial framework for 2-quasi-categories by characterizing them and their fibrations through inner horn inclusions and equivalence extensions, facilitating their study.

## Contribution

It proves that 2-quasi-categories and their fibrations can be characterized using inner horn inclusions and equivalence extensions, simplifying their combinatorial analysis.

## Key findings

- Characterization of 2-quasi-categories via inner horn inclusions
- Simplification of fibrations into 2-quasi-categories
- Provides a combinatorial foundation for further research

## Abstract

Dimitri Ara's 2-quasi-categories, which are certain presheaves over Andr\'{e} Joyal's 2-cell category $\Theta_2$, are an example of a concrete model that realises the abstract notion of $(\infty,2)$-category. In this paper, we prove that the 2-quasi-categories and the fibrations into them can be characterised using the inner horn inclusions and the equivalence extensions introduced by David Oury. These maps are more tractable than the maps that Ara originally used and therefore our result can serve as a combinatorial foundation for the study of 2-quasi-categories.