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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 30th 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: \tableofcontents ## Definition A [[quasicategory]] $C$ is __confluent__ if every [[cospan]] $Q\colon\Lambda^2_0\to C$ the quasicategory $I_{Q/}$ of [[cocones]] under $Q$ is weakly contractible. ## Properties If $C$ has [[pushouts]], then $C$ is confluent because the quasicategory of cocones has an [[initial object]]. A [[quasicategory]] $C$ is confluent if and only if for every morphism $f\colon A\to B$, the induced functor $f^*\colon C_{B/}\to C_{A/}$ is a [[final functor]]. If $\pi\colon T\to B$ is a [[left fibration]] and $B$ is confluent, then so is $T$. \begin{theorem} (Sattler–Wärn.) A quasicategory $C$ is confluent if and only if $C$-indexed [[colimits]] commute with [[pullbacks]] in the quasicategory of [[∞-groupoids]]. \end{theorem} A quasicategory is [[filtered]] if and only if it is confluent and weakly contractible. ## Related concepts * [[confluent category]], [[confluent relation]] * [[filtered colimit]], [[commutativity of limits and colimits]] ## References * [[Christian Sattler]], David Wärn, [A better synthetic argument that confluent colimits commute with pullbacks](https://www.cse.chalmers.se/~sattler/docs/confluent/new-2025.txt), February 2025. Expository account: * [[Rune Haugseng]], Yet another introduction to ∞-categories, 2025. [PDF](https://runegha.folk.ntnu.no/naivecat_web.pdf). <a href="https://ncatlab.org/nlab/revision/confluent+%28%E2%88%9E%2C1%29-category/1">v1</a>, <a href="https://ncatlab.org/nlab/show/confluent+%28%E2%88%9E%2C1%29-category">current</a>

   Created:

   \tableofcontents

   Definition

   A quasicategory $C$ is confluent if every cospan $Q\colon\Lambda^2_0\to C$ the quasicategory $I_{Q/}$ of cocones under $Q$ is weakly contractible.

   Properties

   If $C$ has pushouts, then $C$ is confluent because the quasicategory of cocones has an initial object.

   A quasicategory $C$ is confluent if and only if for every morphism $f\colon A\to B$, the induced functor $f^*\colon C_{B/}\to C_{A/}$ is a final functor.

   If $\pi\colon T\to B$ is a left fibration and $B$ is confluent, then so is $T$.

   \begin{theorem} (Sattler–Wärn.) A quasicategory $C$ is confluent if and only if $C$-indexed colimits commute with pullbacks in the quasicategory of ∞-groupoids. \end{theorem}

   A quasicategory is filtered if and only if it is confluent and weakly contractible.

   Related concepts

   • confluent category, confluent relation

   • filtered colimit, commutativity of limits and colimits

   References

   • Christian Sattler, David Wärn, A better synthetic argument that confluent colimits commute with pullbacks, February 2025.

   Expository account:

   • Rune Haugseng, Yet another introduction to ∞-categories, 2025. PDF.

   v1, current