On local fibrations of $(\infty,2)$-categories

Abstract

In this work we provide a model-independent notion of local fibrations of $(\infty,2)$-categories which generalises the well-known theory of locally coCartesian fibrations of $(\infty,1)$-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 $(\infty,2)$-categories between the $(\infty,2)$-category of $(0,1)$-fibrations over $S$ (also known as inner coCartesian fibrations) and the $(\infty,2)$-category of functors $S \to \mathbb{C}\!\operatorname{at}_{(\infty,2)}$ with values in $(\infty,2)$-categories. Given an $(\infty,2)$-category $\mathbb{B}$, our Grothendieck construction can be specialised to produce an equivalence between the $(\infty,2)$-category of local fibrations over $\mathbb{B}$ and the $(\infty,2)$-category of oplax unital functors with values in $\mathbb{C}\!\operatorname{at}_{(\infty,2)}$. Finally, as an application of our results we provide a version of the Yoneda lemma for $(\infty,2)$-categories.