Dualities in the complicial model of $\infty$-categories

TL;DR

This paper explores dualities and fibrations in the complicial model of infinity-categories, establishing connections between Gray tensor product, suspension, and weak equivalences, and introducing a new co-duality involution.

Contribution

It introduces the co-duality, a weak involution reversing even-dimensional cells, and characterizes weak equivalences as fully faithful and essentially surjective functors.

Findings

Characterization of weak equivalences as fully faithful and essentially surjective

Construction of the co-duality involution reversing even-dimensional cells

Analysis of Grothendieck fibrations in complicial sets

Abstract

In this note, we study the connection between Gray tensor product and suspension. We derive a characterization of weak equivalences as fully faithful and essentially surjective functors. We construct the $co$ duality, a weak involution that reverses the direction of even dimensional cells. We conclude by studying Grothendieck fibrations of complicial sets.