On local fibrations of (∞,2)-categories

In this work we provide a model-independent notion of local fibrations of (, 2) -categories which generalises the well-known theory of locally coCartesian fibrations of (, 1) -categories. Based on previous work, we construct a model category which serves as a specific combinatorial model for this type of fibrations. Our main result is a generalisation of the locally coCartesian straightening and unstraightening construction of Lurie, which yields for any scaled simplicial set S an equivalence of (, 2) -categories between the (, 2) -category of (0, 1) -fibrations over S (also known as inner coCartesian fibrations) and the (, 2) -category of functors S C\!at (, ₂) with values in (, 2) -categories. Given an (, 2) -category B, our Grothendieck construction can be specialised to produce an equivalence between the (, 2) -category of local fibrations over B and the (, 2) -category of oplax unital functors with values in C\!at (, ₂). Finally, as an application of our results we provide a version of the Yoneda lemma for (, 2) -categories.