Fernando Abellán
Given a marked ∞-category (i.e. an ∞-category equipped with a specified collection of morphisms) and a functor with values in an ∞-bicategory, we define marked colimit of F . We provide a definition of weighted colimits in ∞-bicategories when the indexing diagram is an ∞-category and show that they can be computed in terms of marked colimits. In the maximally marked case, our construction retrieves the ∞-categorical colimit of F in the underlying ∞-category. We further relate our construction with the definition of lax colimit of Gepner-Haugseng-Nikolaus. We show that a suitable ∞-localization of the associated coCartesian fibration of a functor with values in ∞-categories computes the marked colimit. Our main theorem is a characterization of those functors of marked ∞-categories which are marked cofinal. More precisely, we provide sufficient and necessary criteria for the restriction of diagrams along f to preserve marked colimits
