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

TL;DR

This paper develops a Yoneda lemma for $ ext{D}$-simplicial spaces by introducing localized $ ext{D}$-left fibrations, establishing a model structure, and applying it to Cartesian fibrations of $( ext{infinity},n)$-categories, generalizing classical categorical results.

Contribution

It defines localized $ ext{D}$-left fibrations, constructs a related model structure, and extends the Yoneda lemma and Grothendieck construction to $( ext{infinity},n)$-categories.

Findings

Established a model structure for localized $ ext{D}$-left fibrations.

Proved a Yoneda lemma for $ ext{D}$-simplicial spaces.

Applied the framework to Cartesian fibrations of $( ext{infinity},n)$-categories.

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.