Rigidification of higher categorical structures

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.