2020/06/22 by García, Fernando Abellán
#Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2006.12416
Given a marked ∞-category D† (i.e. an ∞-category equipped with a specified collection of morphisms) and a functor F: D → \mathbbB with values in an ∞-bicategory, we define colim† F, the 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 D\sharp, our construction retrieves the ∞-categorical colimit of F in the underlying ∞-category B ⊆ \mathbbB. In the specific case when \mathbbB=\mathfrakCat∞, the ∞-bicategory of ∞-categories and D\flat is minimally marked, we recover the definition of lax colimit of Gepner-Haugseng-Nikolaus. We show that a suitable ∞-localization of the associated coCartesian fibration UnD(F) computes colim† F. Our main theorem is a characterization of those functors of marked ∞-categories f:C† → D† which are marked cofinal. More precisely, we provide sufficient and necessary criteria for the restriction of diagrams along f to preserve marked colimits.