[Paper Review] Marked colimits and higher cofinality

Fernando Abellán García|arXiv (Cornell University)|Jun 22, 2020

Homotopy and Cohomology in Algebraic Topology7 references4 citations

TL;DR

This paper introduces marked colimits in ∞-bicategories, generalizing weighted and lax colimits via a marked ∞-category structure that controls laxness. It establishes a 2-dimensional universal property for marked colimits and provides a cofinality criterion for functors between marked ∞-categories, showing that marked cofinal functors preserve marked colimits via localization of coCartesian fibrations.

ABSTRACT

Given a marked $\infty$-category $\mathcal{D}^{\dagger}$ (i.e. an $\infty$-category equipped with a specified collection of morphisms) and a functor $F: \mathcal{D} o \mathbb{B}$ with values in an $\infty$-bicategory, we define $\operatorname{colim}^{\dagger} F$, 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 $\mathcal{D}^{\sharp}$, 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 $\mathcal{D}^{\flat}$ 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 $\operatorname{Un}_{\mathcal{D}}(F)$ computes $\operatorname{colim}^{\dagger} F$. Our main theorem is a characterization of those functors of marked $\infty$-categories $f:\mathcal{C}^{\dagger} o \mathcal{D}^{\dagger}$ which are marked cofinal. More precisely, we provide sufficient and necessary criteria for the restriction of diagrams along $f$ to preserve marked colimits.

Motivation & Objective

• To develop a general theory of weighted colimits in ∞-bicategories using marked ∞-categories.
• To define marked colimits as a unifying framework that recovers both ∞-categorical colimits and lax colimits.
• To provide a characterization of marked cofinal functors that preserve marked colimits.
• To relate the construction to the Grothendieck construction and coCartesian fibrations via localization.
• To establish a 2-dimensional universal property for marked colimits, aligning with higher categorical universal constructions.

Proposed method

• Define marked colimits as a universal construction in ∞-bicategories using a specified collection of marked morphisms in the indexing ∞-category.
• Show that weighted colimits in ∞-bicategories can be computed as marked colimits when the indexing diagram is an ∞-category.
• Construct the marked colimit via localization of the associated coCartesian fibration of the functor F.
• Use the 2-dimensional universal property to characterize marked colimits, ensuring uniqueness and compatibility with higher morphisms.
• Establish a cofinality criterion by analyzing the behavior of functors between marked ∞-categories under restriction.
• Relate the construction to the Grothendieck construction, generalizing known results for lax colimits and ∞-colimits.

Experimental results

Research questions

• RQ1 How can weighted colimits in ∞-bicategories be systematically defined and computed?
• RQ2 In what sense do marked colimits generalize both ∞-categorical colimits and lax colimits?
• RQ3 What conditions ensure that a functor between marked ∞-categories preserves marked colimits?
• RQ4 How does the localization of the coCartesian fibration associated to a functor compute the marked colimit?
• RQ5 What is the precise 2-dimensional universal property satisfied by marked colimits?

Key findings

• Marked colimits recover the standard ∞-categorical colimit when the marking is maximal (i.e., all morphisms are marked).
• In the case of the ∞-bicategory of ∞-categories, with the minimal marking, the marked colimit recovers the lax colimit as defined by Gepner-Haugseng-Nikolaus.
• The marked colimit of a functor F is computed as the localization of the associated coCartesian fibration Un_D(F) at the marked morphisms.
• A functor f: C^† → D^† is marked cofinal if and only if it preserves marked colimits, providing a necessary and sufficient condition.
• The construction satisfies a 2-dimensional universal property, making it suitable for higher categorical universal algebra.
• The theory unifies and generalizes existing results on lax colimits and ∞-colimits through the framework of marked ∞-categories.