Marked colimits and higher cofinality

Abstract Given a marked $$\infty $$ ∞ -category $$\mathcal {D}^{\dagger }$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">D† (i.e. an equipped with specified collection of morphisms) and functor $$F: \mathcal {D}\rightarrow {\mathbb {B}}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">F:D→B values in -bicategory, we define "Equation missing", the colimit F . We provide definition weighted colimits -bicategories when indexing diagram is show that they can be computed terms colimits. In maximally case {D}^{\sharp xmlns:mml="http://www.w3.org/1998/Math/MathML">D♯ , our construction retrieves -categorical underlying {B}\subseteq xmlns:mml="http://www.w3.org/1998/Math/MathML">B⊆B specific -bicategory -categories {D}^{\flat xmlns:mml="http://www.w3.org/1998/Math/MathML">D♭ minimally marked, recover lax Gepner–Haugseng–Nikolaus. suitable -localization associated coCartesian fibration $${\text {Un}}_{\mathcal {D}}(F)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">UnD(F) computes -->. Our main theorem characterization those functors $${f:\mathcal {C}^{\dagger } \rightarrow }}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">f:C†→D† which are cofinal. More precisely, sufficient necessary criteria for restriction diagrams along f to preserve