Marked colimits and higher cofinality

Springer Science and Business Media LLC - Tập 17 Số 1 - Trang 1-22 - 2022
Fernando Abellán García1
1Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146, Hamburg, Germany

Tóm tắt

AbstractGiven a marked$$\infty $$∞-category$$\mathcal {D}^{\dagger }$$D†(i.e. an$$\infty $$∞-category equipped with a specified collection of morphisms) and a functor$$F: \mathcal {D}\rightarrow {\mathbb {B}}$$F:D→Bwith values in an$$\infty $$∞-bicategory, we define "Equation missing", the marked colimit ofF. 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$$\mathcal {D}^{\sharp }$$D♯, our construction retrieves the$$\infty $$∞-categorical colimit ofFin the underlying$$\infty $$∞-category$$\mathcal {B}\subseteq {\mathbb {B}}$$B⊆B. In the specific case when "Equation missing", the$$\infty $$∞-bicategory of$$\infty $$∞-categories and$$\mathcal {D}^{\flat }$$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$${\text {Un}}_{\mathcal {D}}(F)$$UnD(F)computes "Equation missing". Our main theorem is a characterization of those functors of marked$$\infty $$∞-categories$${f:\mathcal {C}^{\dagger } \rightarrow \mathcal {D}^{\dagger }}$$f:C†→D†which are marked cofinal. More precisely, we provide sufficient and necessary criteria for the restriction of diagrams alongfto preserve marked colimits

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Springer Science and Business Media LLC

Tập 17 Số 1

1-22

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