This paper deals with lax colimits depending on a collection of marked morphisms in the source, which can be seen as controlling the laxness of the colimit in question. Such collections of marked morphisms arise throughout the study of higher categories. The theory of marked colimits and marked cofinality presented in this paper represents a new technology. Along the way to \(\infty\)-cofinality, it is shown that the theory weighted colimits expounded in [D. Gepner et al., Doc. Math. 22, 1225–1266 (2017; Zbl 1390.18021)] can be viewed as one instance of the general theory of marked colimits, noting a fundamental relation to the Grothendieck construction generalizing extant results for lax colimits and usual \(\infty\)-colimits.
Given a marked \(\infty\)-category \(\mathcal{D}^{†}\)and a functor \(F:\mathcal{D}\rightarrow\mathbb{B}\)with values in an \(\infty\)-bicategory, this paper defines \(\operatorname{colim}^{†} F\), the marked colimit of \(F\). The author provides a definition of weighted colimits in \(\infty\)-bicategories when the indexing diagram is an \(\infty\)-category, showing that they can be computed in terms of marked colimits. It is shown that a suitable \(\infty\)-localization of the associated coCartesian fibration \(\mathrm{Un}_{\mathcal{D}}(F)\)computes \(\operatorname{colim}^{†} F\). The main result is a characterization of those functors of marked \(\infty\)-categories \(f:\mathcal{C}^{†}\rightarrow\mathcal{D}^{†}\) which are marked cofinal.