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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.
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,
This is the first of a series of papers on enriched infinity categories, seeking to reduce enriched higher category theory to the higher algebra of presentable infinity categories, which is better understood and can be approached via universal proper
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.
Restriction categories were introduced to provide an axiomatic setting for the study of partially defined mappings; they are categories equipped with an operation called restriction which assigns to every morphism an endomorphism of its domain, to be