Joint work with David Wärn.

History:
* 2025-02-19: initial version
* 2025-02-20: moved remarks to end, minor edits

Here is a fibrational perspective on confluent colimits commuting with pullbacks in spaces.
(The fibrational perspective is inspired by [this note from 2020](https://www.cse.chalmers.se/~sattler/docs/confluent/pullbacks.txt), but the new argument is much simpler.)
We work with (by default, higher) categories in a way that abstracts over the concrete model.

The setting
===========

Definition.
A category C is *confluent* if the category of cocones under any span has terminal localization.
∎

That is, the constant functor C → [{• ← • → •}, C] is final.
We wish to show that confluent colimits commute with pullbacks in spaces.
This decomposes the sound doctrine of filtered colimits into the sound doctrines of confluent colimits and ∞-connected colimits.
In particular, a category is filtered exactly if it is confluent and has terminal localization ("∞-connected").
Confluence has some nice closure properties that filteredness lacks, for example Corollary 2 below.

Background (factorization systems for left and right fibrations).
Recall the following factorization systems in categories:
* The functor {0} ⊆ [1] generates the factorization system (*initial*, *left fibration*).
* The functor {1} ⊆ [1] generates the factorization system (*final*, *right fibration*).
These are reciprocally Frobenius:
* pullback along left fibrations preserves final functors,
* pullback along right fibrations preserves initial functors.
This can be seen by reducing to the case of the generator.

Given a category C, the factorization systems induce reflectors L_left and L_right for the inclusions of left and right fibrations with base C into categories over C admit reflectors .
By reciprocal Frobenius:
* reflection to left fibrations is stable under base change along right fibrations,
* reflection to right fibrations is stable under base change along left fibrations.
∎

Background (free right fibrations via free cartesian fibrations).
We may implement L_right(π : E → C) by taking the free cartesian fibration
  C ↓ π ---> C
and localizing fiberwise (and dually for L_left).
If E is a groupoid, the second step can be skipped as the fibers of C ↓ π are already groupoids.
In particular, the right fibration reflection of an object selector x : 1 → C is given by the projection C ↓ x → C.
∎

Background (Kan fibrations).
A *Kan fibration* is a functor that is both a left and a right fibration.
This is the right class in the factorization system generated by {0} ⊆ [1] and {1} ⊆ [1].
[The left class is given by functors that become invertible under localization.]

By descent for the localization of C, the reflective embedding
  KanFib(C) ---> Cat ↓ C
corresponds to the reflective embedding
  Space ↓ |C| ---> Cat ↓ C
given by pullback along C → |C|, with reflector given by localization.
Furthermore, this correspondence is natural in C (with functorial action given by base change).
∎

The argument
============

Lemma 1.
The following are equivalent:
(1) C is confluent,
(2) for f : x → y, postcomposition f^! : y ↓ C → x ↓ C is final,

Proof.
Recall that f^! is final exactly if (x', g) ↓ f^! has terminal realization for g : x → x'.
But (x', g) ↓ f^! is the category of cocones under the span (f, g).
And (1) means that this category has terminal realization.
∎

Corollary 2.
Consider a left fibration E → C.
If C is confluent, then so is E.

This can be shown directly using that the walking span has an initial object.
But we will deduce it from Lemma 1 in a symmetry-breaking way that the main argument will mirror.
Note that the case C = 1 says that every groupoid is confluent.

Proof.
Consider a map g : a → b of E over a map f : x → y of C.
This induces the square of categories
  b ↓ E ---> a ↓ E
  |          |
  y ↓ C ---> x ↓ C.
Since the functor E → C is a left fibration, the vertical functors are invertible.
Therefore, the top functor is final exactly if the bottom functor is.
The claim follows using the characterization of confluence given by Lemma 1.
∎

Lemma 3.
Assume that C is confluent.
In categories over C, reflection to right fibrations preserves left fibrations.

Proof.
Consider a left fibration E → C.
The reflection L_right(E) → C is given by:
  L_right(E)(x) = colim E|_{x ↓ C}
with functorial action f : x → y given by the map
  colim E|_{x ↓ C} <--- colim E|_{y ↓ C}
induced by restricting the colimit shape along f^!.
This is invertible by the previous lemma, making L_right(E) a left fibration.
∎

Remark.
Lemma 3 expresses a compatibility between the modalities L_left and L_right.
Since L_right preserves L_left-modal objects, the intersection L_Kan of the modalities is given by L_right L_left.
Note that if C is both confluent and coconfluent, then L_left and L_right commute.
∎

Corollary 4.
Consider a left fibration F : C' → C with C confluent.
Base change along F preserves reflection of left fibrations to Kan fibrations.

Proof.
Note that C' is also confluent by Corollary 2.
By Lemma 3, reflection to Kan fibrations agrees with reflection to right fibrations for bases C and C'.
And reflection to right fibrations is stable under base change along F since F is a left fibration.
∎

Corollary 5.
In spaces, confluent colimits commute with pullbacks.

Proof.
Consider a pullback
  Y' ---> Y
  | pb    |
  X' ---> X
of left fibrations over a confluent category C.
We must show that the square of localizations
  |Y'| ---> |Y|
  |         |
  |X'| ---> |X|
is a pullback.
By the descent statement in the background on Kan fibrations, this equivalently means the square
  L_Kan(Y') ---> L_Kan(Y)
  |              |
  X' ----------> X
is a pullback.
By Corollary 2, X is confluent.
So the goal holds by Corollary 4.
∎

Remarks
=======

Background (optional, coreflection of left and right fibrations).
Left and right fibrations with base C also admit coreflectors R_left and R_right.
In particular, the inclusions of left and right fibrations into categories over C preserve colimits.

We explain one way to see this, for the case of right fibrations.
Simplicial spaces have analogous factorization systems for right fibrations, generated by terminal object inclusions Δ[0] → Δ[n].
The generators goes between representables, so mapping out of their source and target preserves colimits.
So by universality of colimits, right fibrations with base X are closed under colimits.
[Remark: if we remove the requirement that the base is fixed, this is still true for confluent colimits.]
Therefore, the inclusion of right fibrations with base X into simplicial spaces over X admits a right adjoint R_right.

Categories embed into simplicial spaces via the nerve N.
This embedding creates right fibrations.
So the right adjoint R_right descends from simplicial spaces over N C to categories over C.
For example, the objects of R_right(π : E → C) over x : C are given by functors C ↓ x → E over C.
∎

Remark.
Dually to Lemma 3, one may show that, over a confluent base, coreflection to left fibrations preserves right fibrations.
∎

Remark (L_Kan in terms of L_left and L_right).
Recall that left and right fibrations are closed under sequential colimits.
So we may implement L_Kan using the generalized zig-zag construction as a sequential colimit of:
  Id ---> L_left ---> L_right L_left ---> L_left L_right L_left ---> …
and dually with left and right exchanged.

Lemma 3 then says that this construction terminates after two steps (when started with L_left).
(And it terminates after three steps when started with L_right.)
∎