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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.
We present a version of enriched Yoneda lemma for conventional (not infinity-) categories. We require the base monoidal category to have colimits, but do not require it to be closed or symmetric monoidal.
We continue the study of enriched infinity categories, using a definition equivalent to that of Gepner and Haugseng. In our approach enriched infinity categories are associative monoids in an especially designed monoidal category of enriched quivers.
We develop some basic concepts in the theory of higher categories internal to an arbitrary $infty$-topos. We define internal left and right fibrations and prove a version of the Grothendieck construction and of Yonedas lemma for internal categories.
We prove the uniqueness, the functoriality and the naturality of cylinder objects and path objects in closed simplicial model categories.
Many special classes of simplicial sets, such as the nerves of categories or groupoids, the 2-Segal sets of Dyckerhoff and Kapranov, and the (discrete) decomposition spaces of G{a}lvez, Kock, and Tonks, are characterized by the property of sending ce