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

给定一个标记的\(\infty \)类别\(\mathcal {D}^{\dagger }\)（即配备有指定态射集合的\(\infty \)类别）和一个函子\(F : \mathcal {D}\rightarrow {\mathbb {B}}\)的值在\(\infty \) -bicategory 中，我们定义 ， F的标记上界。当索引图是\ (\infty \) 类别时，我们提供了 \ (\infty \)双类别中的加权余极限的定义，并表明它们可以根据标记的余极限来计算。在最大标记情况\(\mathcal {D}^{\sharp }\)中，我们的构造检索底层\(\infty \)类别\( \ mathcal 中F的 \ (\infty \)分类余极限{B}\subseteq {\mathbb {B}}\)。在特定情况下，当， \(\infty \ )类别和\(\mathcal {D}^{\flat }\)的\(\infty \) 双类别被最小化标记时，我们恢复宽松余极限的定义格普纳-豪格森-尼古拉斯。我们证明了相关协笛卡尔纤维的合适的\(\infty \)定位\({\text {Un}}_{\mathcal {D}}(F)\)计算。我们的主要定理是标记\(\infty \)类别\({f:\mathcal {C}^{\dagger } \rightarrow \mathcal {D}^{\dagger }}\)的函子的特征，其中被标记为 cofinal。更准确地说，我们为沿f的图的限制提供充分且必要的标准，以保留标记的余界