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

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 نشر من قبل Nima Rasekh
 تاريخ النشر 2021
  مجال البحث
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 تأليف Nima Rasekh




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For a small category $mathcal{D}$ we define fibrations of simplicial presheaves on the category $mathcal{D}timesDelta$, 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.



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