Marked colimits and higher cofinality

Abstract

Given a marked \(\infty \)-category \(\mathcal {D}^{\dagger }\) (i.e. an \(\infty \)-category equipped with a specified collection of morphisms) and a functor \(F: \mathcal {D}\rightarrow {\mathbb {B}}\) with values in an \(\infty \)-bicategory, we define , the marked colimit of F. We provide a definition of weighted colimits in \(\infty \)-bicategories when the indexing diagram is an \(\infty \)-category and show that they can be computed in terms of marked colimits. In the maximally marked case \(\mathcal {D}^{\sharp }\), our construction retrieves the \(\infty \)-categorical colimit of F in the underlying \(\infty \)-category \(\mathcal {B}\subseteq {\mathbb {B}}\). In the specific case when , the \(\infty \)-bicategory of \(\infty \)-categories and \(\mathcal {D}^{\flat }\) is minimally marked, we recover the definition of lax colimit of Gepner–Haugseng–Nikolaus. We show that a suitable \(\infty \)-localization of the associated coCartesian fibration \({\text {Un}}_{\mathcal {D}}(F)\) computes . Our main theorem is a characterization of those functors of marked \(\infty \)-categories \({f:\mathcal {C}^{\dagger } \rightarrow \mathcal {D}^{\dagger }}\) which are marked cofinal. More precisely, we provide sufficient and necessary criteria for the restriction of diagrams along f to preserve marked colimits