Homotopy theory of higher categories

TL;DR

This paper develops a homotopy-theoretic framework for higher categories by constructing model structures on categories of precategories, enabling iterative modeling of (, n)-categories using Segal's method.

Contribution

It introduces a new model structure on M-precategories for higher categories, extending to (, n)-categories through iteration, based on Segal's approach.

Findings

Constructed a tractable model structure on M-precategories.

Enabled iterative modeling of (, n)-categories.

Provided a foundation for homotopy theory of higher categories.

Abstract

This is the first draft of a book about higher categories approached by iterating Segal's method, as in Tamsamani's definition of $n$-nerve and Pelissier's thesis. If $M$ is a tractable left proper cartesian model category, we construct a tractable left proper cartesian model structure on the category of $M$-precategories. The procedure can then be iterated, leading to model categories of $(\infty, n)$-categories.