显着的共极限和更高的共尾性
Marked colimits and higher cofinality
摘要：
给定一个带标记的$\infty$ -范畴$\mathcal{D}^{\dagger}$（即一个带有指定态射集合的$\infty$ -范畴）和一个值在$\infty$ -双范畴中的函子$F: \mathcal{D} \to \mathbb{B}$，我们定义了$F$的带标记余极限$\operatorname{colim}^{\dagger} F$.我们提供了一个定义的加权共极限在$\infty$ -bicategories时的索引图是一个$\infty$ -类别，并表明，它们可以计算的标记共极限。在最大标记的情况下$\mathcal{D}^{\sharp}$，我们的构造检索$\infty$-范畴$\mathcal{B} \subseteq \mathbb{B}$中的$F$的$\infty$ -范畴余限。当$\mathbb{B}=\mathfrak{Cat}_{\infty}$、$\infty$ -范畴和$\mathcal{D}^{\flat}$的$\infty$ -双范畴是极小标记的特殊情况下，我们恢复了Gepner-Haugseng-Nikolaus的松弛余限的定义。我们表明，一个合适的$\infty$ -本地化的相关coCartesian纤维化$\operatorname{Un}_{\mathcal{D}}(F)$计算$\operatorname{colim}^{\dagger} F$。我们的主要定理是对标记$\infty$ -范畴$f:\mathcal{C}^{\dagger} \to \mathcal{D}^{\dagger}$中标记为共尾的函子的一个刻画.更确切地说，我们提供了充分和必要的标准，限制图沿$f$保存标记colimits。
英文摘要：
Given a marked $\infty$-category $\mathcal{D}^{\dagger}$ (i.e. an $\infty$-category equipped with a specified collection of morphisms) and a functor $F: \mathcal{D} \to \mathbb{B}$ 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 $f:\mathcal{C}^{\dagger} \to \mathcal{D}^{\dagger}$ which are marked cofinal. More precisely, we provide sufficient and necessary criteria for the restriction of diagrams along $f$ to preserve marked colimits.
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