Yoneda's lemma for internal higher categories

TL;DR

This paper extends fundamental concepts of higher category theory, including Yoneda's lemma, to the setting of internal categories within an arbitrary infinity-topos, broadening the theoretical framework.

Contribution

It introduces internal left and right fibrations and proves a version of the Grothendieck construction and Yoneda's lemma for internal higher categories.

Findings

Established internal left and right fibrations in higher categories

Proved a version of the Grothendieck construction for internal categories

Extended Yoneda's lemma to internal higher categories

Abstract

We develop some basic concepts in the theory of higher categories internal to an arbitrary $\infty$-topos. We define internal left and right fibrations and prove a version of the Grothendieck construction and of Yoneda's lemma for internal categories.