2020/12/08 by Gagna, Andrea, Harpaz, Yonatan, Lanari, Edoardo
#18N50 #18N65 #55U35 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2012.04537
We study four types of (co)cartesian fibrations of ∞-bicategories over a given base B, and prove that they encode the four variance flavors of B-indexed diagrams of ∞-categories. We then use this machinery to set up a general theory of 2-(co)limits for diagrams valued in an ∞-bicategory, capable of expressing lax, weighted and pseudo limits. When the ∞-bicategory at hand arises from a model category tensored over marked simplicial sets, we show that this notion of 2-(co)limit can be calculated as a suitable form of a weighted homotopy limit on the model categorical level, thus showing in particular the existence of these 2-(co)limits in a wide range of examples. We finish by discussing a notion of cofinality appropriate to this setting and use it to deduce the unicity of 2-(co)limits, once exist.