Some properties of the theory of n-categories

TL;DR

This paper explores the properties of the Dwyer-Kan localization of weak n-categories, proposing conjectural characterizations and methods to derive morphism categories and compositions within this framework.

Contribution

It introduces a set of properties conjecturally characterizing the localization of weak n-categories and demonstrates how to extract morphism categories and composition maps from these properties.

Findings

Proposes properties that characterize the localization up to equivalence

Shows how to obtain morphism n-1 categories between points

Demonstrates how to derive composition maps between morphism objects

Abstract

Let $L_n$ denote the Dwyer-Kan localization of the category of weak n-categories divided by the n-equivalences. We propose a list of properties that this simplicial category is likely to have, and conjecture that these properties characterize $L_n$ up to equivalence. We show, using these properties, how to obtain the morphism $n-1$-categories between two points in an object of $L_n$ and how to obtain the composition map between the morphism objects.