On the equivalence between $\Theta_{n}$-spaces and iterated Segal spaces

TL;DR

This paper provides a new proof demonstrating the equivalence between two primary models for $( abla,n)$-categories, showing they are algebras for the same monad on $n$-globular spaces, applicable to a wide class of $ abla$-categories.

Contribution

The paper introduces a novel proof establishing the equivalence between $n$-fold Segal spaces and $ abla_{n}$-spaces via monad algebras, broadening the understanding of $( abla,n)$-categories.

Findings

Proves the equivalence of two models for $( abla,n)$-categories.

Shows both models are algebras for the same monad.

Applicable to all $ abla$-categories including $ abla$-topoi.

Abstract

We give a new proof of the equivalence between two of the main models for $(\infty,n)$-categories, namely the $n$-fold Segal spaces of Barwick and the $\Theta_{n}$-spaces of Rezk, by proving that these are algebras for the same monad on the $\infty$-category of $n$-globular spaces. The proof works for a broad class of $\infty$-categories that includes all $\infty$-topoi.