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    Understanding higher structures through Quillen-Segal objects

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    If M\mathscr{M} is a model category and U:A→M\mathcal{U}: \mathscr{A} \rightarrow \mathscr{M} is a functor, we defined a Quillen-Segal U\mathcal{U}-object as a weak equivalence F:s(F)→∼t(F)\mathscr{F}: s(\mathscr{F}) \xrightarrow{\sim} t(\mathscr{F}) such that t(F)=U(b)t(\mathscr{F})=\mathcal{U}(b) for some b∈Ab\in \mathscr{A}. If U\mathcal{U} is the nerve functor U:Cat→sSetJ\mathcal{U}: \mathbf{Cat} \rightarrow \mathbf{sSet}_J, with the Joyal model structure on sSet\mathbf{sSet}, then studying the comma category (sSetJ↓U)(\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} U:sCatB→sSetJ\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 U\mathcal{U} is the identity Id:sSetQ→sSetQId: \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 M\mathscr{M}. This result agrees with our intuition, since by essence the `\textit{Quillen-Segal type}' is the \textit{Equivalence type}Comment: 9 pages, First draft. Comment are always welcom
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