Marked colimits and higher cofinality

Abstract

Given a marked $\infty$-category ${\mathcal{D}}^{†}$ (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 ${\mathrm{colim}}^{†}F$, 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 ${\mathcal{D}}^{♯}$, 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 ${\mathcal{D}}^{♭}$ 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 ${\mathrm{Un}}_{\mathcal{D}}(F)$ computes ${\mathrm{colim}}^{†}F$. Our main theorem is a characterization of those functors of marked $\infty$-categories $f:{\mathcal{C}}^{†}\to {\mathcal{D}}^{†}$ which are marked cofinal. More precisely, we provide sufficient and necessary criteria for the restriction of diagrams along $f$ to preserve marked colimits.

17 pages. Minor revisions. Submitted for publication. Comments welcome