An $(\infty,n)$-categorical straightening-unstraightening construction

Abstract

We provide an $(\infty,n)$-categorical version of the straightening-unstraightening construction, asserting an equivalence between the $(\infty,n)$-category of double $(\infty,n-1)$-right fibrations over an $(\infty,n)$-category $\mathcal{C}$ and that of the $(\infty,n)$-functors from $\mathcal{C}$ valued in $(\infty,n-1)$-categories. We realize this in the form of a Quillen equivalence between appropriate model structures; on the one hand, a model structure for double $(\infty,n-1)$-right fibrations over a generic precategory object $W$ in $(\infty,n-1)$-categories and, on the other hand, a model structure for $(\infty,n)$-functors from its homotopy coherent categorification $\mathfrak{C} W$ valued in $(\infty,n-1)$-categories.