Understanding higher structures through Quillen-Segal objects

Hugo Bacard (1)

1 Department of Mathematics [London, Ontario]

If $\M$ is a model category and $\Ub: \ag \to \M$ is a functor, we defined a Quillen-Segal $\Ub$-object as a weak equivalence $\F: s(\F) \xrightarrow{\sim} t(\F)$ such that $t(\F)=\Ub(b)$ for some $b\in \ag$. If $\Ub$ is the nerve functor $\Ub: \Cat \to ßetj$, with the Joyal model structure on $ßet$, then studying the comma category $(ßetj \downarrow \Ub)$ 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 \emph{coherent nerve} $\Ub: \sCatb \to ßetj$, then the theory of QS-objects directly connects with the program of Riehl and Verity. If we apply our main result when $\Ub$ is the identity $\Id: ßetq \to ßetq$, 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 \emph{avatar} of Voevodsky's \emph{Univalence axiom}. This equivalence holds for any combinatorial and left proper $\M$. This result agrees with our intuition, since by essence the '\emph{Quillen-Segal type}' is the \emph{Equivalence type}

Mots clés: quasicategories; topos; type theory; model categories; Quillen-Segal; co-Segal

Domaines: Catégories et ensembles [math.CT]; Géométrie algébrique [math.AG]; K-théorie et homologie [math.KT]; Topologie générale [math.GN]; Algèbre commutative [math.AC]

Hugo Bacard. Understanding higher structures through Quillen-Segal objects. 2014. ⟨hal-01018216⟩