nLab confluent (∞,1)-category

Definition

A quasicategory $C$ is confluent if for 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 $T$.

Theorem

(Sattler & Wärn 2025) A quasicategory $C$ is confluent if and only if $C$-indexed colimits commute with pullbacks in the quasicategory of ∞-groupoids.

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]