Marked colimits and higher cofinality

Given a marked $\infty$-category $\mathcal{D}^{\dagger}$ (i.e. an $\infty$-category equipped with a specified collection of morphisms) and a functor $F: \mathcal{D} \to \mathbb{B}$ with values in an $\infty$-bicategory, we define $\operatorname{colim}^{\dagger} F$, 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 $\mathbb{B}=\mathfrak{Cat}_{\infty}$, 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 $\operatorname{Un}_{\mathcal{D}}(F)$ computes $\operatorname{colim}^{\dagger} F$. Our main theorem is a characterization of those functors of marked $\infty$-categories $f:\mathcal{C}^{\dagger} \to \mathcal{D}^{\dagger}$ which are marked cofinal. More precisely, we provide sufficient and necessary criteria for the restriction of diagrams along $f$ to preserve marked colimits.