Yoneda lemma for enriched infinity categories

Abstract

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 prove that, in case the monoidal structure in the basic category M comes from direct product, our definition is essentially equivalent to the approach via Segal objects. Furthermore, we compare our notion with the notion of category left-tensored over M, and prove a version of Yoneda lemma in this context. Version 2: An error in 2.6.2 corrected. Version 3: a few minor corrections. Version 4: Section 8 added, describing correspondences of enriched categories. In case the basic monoidal category M is a prototopos with a cartesian structure, we prove that the category of correspondences is equivalent to the category of enriched categories over [1]. Version 5: terminology changed (former bicartesian fibrations became bifibrations), a few misprints corrected. Version 6: Section 2.11 added, dealing with operadic sieves. A number of corrections and clarifications made per referee's request. Version 7: final version, accepted to Advances in Math. Version 8: a minor correction of 2.8.9-2.8.10.