Rigidification of higher categorical structures

TL;DR

This paper establishes a method to rigidify higher categorical structures, showing that 'up to homotopy' models can be replaced by strict models in a way that preserves their homotopical properties, with applications to various models of higher categories.

Contribution

It introduces a general transfer technique for model structures from 'up to homotopy' to strict models, leading to new rigidification theorems for higher categorical structures.

Findings

Rigidification of $ heta_n$-spaces to satisfy Segal conditions strictly.

Transfer of model structures preserves homotopical equivalences.

Applicable to dendroidal and $n$-fold Segal spaces.

Abstract

Given a limit sketch in which the cones have a finite connected base, we show that a model structure of "up to homotopy" models for this limit sketch in a suitable model category can be transferred to a Quillen equivalent model structure on the category of strict models. As a corollary of our general result, we obtain a rigidification theorem which asserts in particular that any $\Theta_n$-space in the sense of Rezk is levelwise equivalent to one that satisfies the Segal conditions on the nose. There are similar results for dendroidal spaces and $n$-fold Segal spaces.