$(\infty,n)$-Limits II: Comparison across models

TL;DR

This paper demonstrates the equivalence of two approaches to defining $( abla,n)$-limits, constructs various double $( abla,n-1)$-categories, and proves key examples are (co)complete, advancing the understanding of higher category theory.

Contribution

It establishes the equivalence of enriched and internal approaches to $( abla,n)$-limits and provides explicit constructions and properties of related higher categories.

Findings

Enriched and internal definitions of $( abla,n)$-limits coincide.

Constructed explicit double $( abla,n-1)$-categories for join, slice, and cone constructions.

Proved key $( abla,n)$-categories are (co)complete.

Abstract

We show that the notion of $(\infty,n)$-limit defined using the enriched approach and the one defined using the internal approach coincide. We also give explicit constructions of various double $(\infty,n-1)$-categories implementing various join constructions, slice constructions and cone constructions, and study their properties. We further prove that key examples of $(\infty,n)$-categories are (co)complete.