Understanding higher structures through Quillen-Segal objects

Abstract

If $\mathscr{M}$ is a model category and $\mathcal{U}: \mathscr{A} \rightarrow \mathscr{M}$ is a functor, we defined a Quillen-Segal $\mathcal{U}$-object as a weak equivalence $\mathscr{F}: s(\mathscr{F}) \xrightarrow{\sim} t(\mathscr{F})$ such that $t(\mathscr{F})=\mathcal{U}(b)$ for some $b\in \mathscr{A}$. If $\mathcal{U}$ is the nerve functor $\mathcal{U}: \mathbf{Cat} \rightarrow \mathbf{sSet}_J$, with the Joyal model structure on $\mathbf{sSet}$, then studying the comma category $(\mathbf{sSet}_J \downarrow \mathcal{U})$ leads naturally to concepts, such as Lurie's $\infty$-operad. It also gives simple examples of presentable, stable $\infty$-category, and higher topos. If we consider the \textit{coherent nerve} $\mathcal{U}: \mathbf{sCat}_B \rightarrow \mathbf{sSet}_J$, then the theory of QS-objects directly connects with the program of Riehl and Verity. If we apply our main result when $\mathcal{U}$ is the identity $Id: \mathbf{sSet}_Q \rightarrow \mathbf{sSet}_Q$, with the Quillen model structure, the homotopy theory of QS-objects is equivalent to that of Kan complexes and we believe that this is an \textit{avatar} of Voevodsky's \textit{Univalence axiom}. This equivalence holds for any combinatorial and left proper $\mathscr{M}$. This result agrees with our intuition, since by essence the `\textit{Quillen-Segal type}' is the \textit{Equivalence type}