Yoneda Lemma for $\mathcal{D}$-Simplicial Spaces

Abstract

For a small category $\mathcal{D}$ we define fibrations of simplicial presheaves on the category $\mathcal{D}\times\Delta$, which we call localized $\mathcal{D}$-left fibration. We show these fibrations can be seen as fibrant objects in a model structure, the localized $\mathcal{D}$-covariant model structure, that is Quillen equivalent to a category of functors valued in simplicial presheaves on $\mathcal{D}$, where the Quillen equivalence is given via a generalization of the Grothendieck construction. We use our understanding of this construction to give a detailed characterization of fibrations and weak equivalences in this model structure and in particular obtain a Yoneda lemma. We apply this general framework to study Cartesian fibrations of $(\infty,n)$-categories, for models of $(\infty,n)$-categories that arise via simplicial presheaves, such as $n$-fold complete Segal spaces. This, in particular, results in the Yoneda lemma and Grothendieck construction for Cartesian fibrations of $(\infty,n)$-categories.