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Understanding higher structures through Quillen-Segal objects
If is a model category and is a functor, we defined a Quillen-Segal
-object as a weak equivalence such that
for some . If is the nerve functor
, with the Joyal model
structure on , then studying the comma category
leads naturally to concepts, such as
Lurie's -operad. It also gives simple examples of presentable, stable
-category, and higher topos. If we consider the \textit{coherent nerve}
, then the theory of
QS-objects directly connects with the program of Riehl and Verity. If we apply
our main result when is the identity , 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 . 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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