feat(FiberedCategory/HasFibers): define HasFibers class

Mathlib.lean

@@ -1549,6 +1549,7 @@ import Mathlib.CategoryTheory.FiberedCategory.Cartesian
 import Mathlib.CategoryTheory.FiberedCategory.Cocartesian
 import Mathlib.CategoryTheory.FiberedCategory.Fiber
 import Mathlib.CategoryTheory.FiberedCategory.Fibered
+import Mathlib.CategoryTheory.FiberedCategory.HasFibers
 import Mathlib.CategoryTheory.FiberedCategory.HomLift
 import Mathlib.CategoryTheory.Filtered.Basic
 import Mathlib.CategoryTheory.Filtered.Connected

Mathlib/CategoryTheory/FiberedCategory/Cartesian.lean

@@ -117,7 +117,18 @@ lemma map_self : IsCartesian.map p f φ φ = 𝟙 a := by
   apply map_uniq
   simp only [id_comp]
 
-/-- The canonical isomorphism between the domains of two cartesian morphisms
+instance of_comp_iso {b' : 𝒳} (φ' : b ≅ b') [IsHomLift p (𝟙 S) φ'.hom] :
+    IsCartesian p f (φ ≫ φ'.hom) where
+  universal_property := by
+    intro c ψ hψ
+    use IsCartesian.map p f φ (ψ ≫ φ'.inv)
+    refine ⟨⟨inferInstance, by simp only [fac_assoc, assoc, Iso.inv_hom_id, comp_id]⟩, ?_⟩
+    rintro τ ⟨hτ₁, hτ₂⟩
+    apply map_uniq
+    rw [Iso.eq_comp_inv]
+    simp only [assoc, hτ₂]
+
+/-- The canonical isomorphism between the domains of two cartesian arrows
 lying over the same object. -/
 @[simps]
 noncomputable def domainUniqueUpToIso {a' : 𝒳} (φ' : a' ⟶ b) [IsCartesian p f φ'] : a' ≅ a where
@@ -152,17 +163,6 @@ instance of_iso_comp {a' : 𝒳} (φ' : a' ≅ a) [IsHomLift p (𝟙 R) φ'.hom]
     apply map_uniq
     simp only [assoc, hτ₂]
 
-/-- Postcomposing a cartesian morphism with an isomorphism lifting the identity is cartesian. -/
-instance of_comp_iso {b' : 𝒳} (φ' : b ≅ b') [IsHomLift p (𝟙 S) φ'.hom] :
-    IsCartesian p f (φ ≫ φ'.hom) where
-  universal_property := by
-    intro c ψ hψ
-    use IsCartesian.map p f φ (ψ ≫ φ'.inv)
-    refine ⟨⟨inferInstance, by simp⟩, ?_⟩
-    rintro τ ⟨hτ₁, hτ₂⟩
-    apply map_uniq
-    simp only [Iso.eq_comp_inv, assoc, hτ₂]
-
 end IsCartesian
 
 namespace IsStronglyCartesian

Mathlib/CategoryTheory/FiberedCategory/HasFibers.lean

@@ -0,0 +1,248 @@
+/-
+Copyright (c) 2024 Calle Sönne. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Calle Sönne, Paul Lezeau
+-/
+
+import Mathlib.CategoryTheory.FiberedCategory.Fiber
+import Mathlib.CategoryTheory.FiberedCategory.Fibered
+
+/-!
+
+# Fibers of functors
+
+In this file we introduce a typeclass `HasFibers` for a functor `p : 𝒳 ⥤ 𝒮`, consisting of:
+- A collection of categories `Fib S` for every `S` in `𝒮` (the fiber categories)
+- Functors `ι : Fib S ⥤ 𝒳` such that `ι ⋙ p = const (Fib S) S
+- The induced functor `Fib S ⥤ Fiber p S` is an equivalence.
+
+We also provide a default `HasFibers` instance, which uses the standard fibers `Fiber p S`
+(see Fiber.lean). This makes it so that any result proven about `HasFibers` can be used for the
+standard fibers as well.
+
+The reason for introducing this typeclass is that in practice, when working with (pre)fibered
+categories one often already has a collection of categories `Fib S` for every `S` that are
+equivalent to the fibers `Fiber p S`. One would then like to use these categories `Fib S` directly,
+instead of working through this equivalence of categories. By developing an API for the `HasFibers`
+typeclass, this will be possible.
+
+Here is an example of when this typeclass is useful. Suppose we have a presheaf of types
+`F : 𝒮ᵒᵖ ⥤ Type _`. The associated fibered category then has objects `(S, a)` where `S : 𝒮` and `a`
+is an element of `F(S)`. The fiber category `Fiber p S` is then equivalent to the discrete category
+`Fib S` with objects `a` in `F(S)`. In this case, the `HasFibers` instance is given by the
+categories `F(S)` and the functor `ι` sends `a : F(S)` to `(S, a)` in the fibered category. See
+`Presheaf.lean` for more details.
+
+## Main API
+The following API is developed so that the fibers from a `HasFibers` instance can be used
+analogously to the standard fibers.
+
+- `mapPreimage φ` is a lift of a morphism `φ : (ι S).obj a ⟶ (ι S).obj b` in `𝒳`, which lies over
+`𝟙 S`, to a morphism in the fiber over `S`.
+- `objPreimage` gives an object in the fiber over `S` which is isomorphic to a given `a : 𝒳` that
+satisfies `p(a) = S`. The isomorphism is given by `objObjPreimageIso`.
+- `HasFibers.pullbackObj` is a version of `IsPreFibered.pullbackObj` which ensures that the object
+lies in a given fiber. The corresponding cartesian morphism is given by `HasFibers.pullbackMap`.
+- `HasFibers.inducedMap` is a version of `IsCartesian.inducedMap` which gives the corresponding
+morphism in the fiber category.
+- `fiber_factorization` is the statement that any morphism in `𝒳` can be factored as a morphism in
+some fiber followed by a pullback.
+
+-/
+
+universe v₁ u₁ v₂ u₂ v₃ u₃
+
+open CategoryTheory Functor Category IsCartesian IsHomLift Fiber
+
+variable {𝒮 : Type u₁} {𝒳 : Type u₂} [Category.{v₁} 𝒮] [Category.{v₂} 𝒳]
+
+/-- HasFibers is an extrinsic notion of fibers on a functor `p : 𝒳 ⥤ 𝒮`. It is given by a
+collection of categories `Fib S` for every `S : 𝒮` (the fiber categories), each equiped with a
+functors `ι : Fib S ⥤ 𝒳` which map constantly to `S` on the base such that the induced functor
+`Fib S ⥤ Fiber p S` is an equivalence. -/
+@[nolint checkUnivs]
+class HasFibers (p : 𝒳 ⥤ 𝒮) where
+  /-- The type of objects of the category `Fib S` for each `S`. -/
+  Fib (S : 𝒮) : Type u₃
+  /-- `Fib S` is a category. -/
+  isCategory (S : 𝒮) : Category.{v₃} (Fib S) := by infer_instance
+  /-- The functor `ι : Fib S ⥤ 𝒳`. -/
+  ι (S : 𝒮) : (Fib S) ⥤ 𝒳
+  /-- The composition with the functor `p` is *equal* to the constant functor mapping to `S`. -/
+  comp_const (S : 𝒮) : (ι S) ⋙ p = (const (Fib S)).obj S
+  /-- The induced functor from `Fib S` to the fiber of `𝒳 ⥤ 𝒮` over `S` is an equivalence. -/
+  equiv (S : 𝒮) : Functor.IsEquivalence (inducedFunctor (comp_const S))
+
+namespace HasFibers
+
+/-- The `HasFibers` on `p : 𝒳 ⥤ 𝒮` given by the fibers of `p` -/
+@[default_instance]
+instance canonical (p : 𝒳 ⥤ 𝒮) : HasFibers p where
+  Fib := Fiber p
+  ι S := @fiberInclusion _ _ _ _ p S
+  comp_const S := fiberInclusion_comp_eq_const
+  equiv S := by exact isEquivalence_of_iso (F := 𝟭 (Fiber p S)) (Iso.refl _)
+
+section
+
+variable (p : 𝒳 ⥤ 𝒮) [HasFibers p] (S : 𝒮)
+
+instance : Category (Fib p S) := isCategory S
+
+/-- The induced functor from `Fib p S` to the standard fiber. -/
+@[simps!]
+def inducedFunctor : Fib p S ⥤ Fiber p S :=
+  Fiber.inducedFunctor (comp_const S)
+
+/-- The natural transformation `ι S ≅ (inducedFunctor p S) ⋙ (fiberInclusion p S)` -/
+def inducedFunctor.NatIso : ι S ≅ (inducedFunctor p S) ⋙ fiberInclusion :=
+  Fiber.inducedFunctorCompIsoSelf (comp_const S)
+
+lemma inducedFunctor_comp : ι S = (inducedFunctor p S) ⋙ fiberInclusion :=
+  Fiber.inducedFunctor_comp (comp_const S)
+
+instance : Functor.IsEquivalence (inducedFunctor p S) := equiv S
+
+instance : Functor.Faithful (ι (p:=p) S) :=
+  Functor.Faithful.of_iso (inducedFunctor.NatIso p S).symm
+
+end
+
+section
+
+variable {p : 𝒳 ⥤ 𝒮} [HasFibers p]
+
+@[simp]
+lemma proj_eq {S : 𝒮} (a : Fib p S) : p.obj ((ι S).obj a) = S := by
+  simp only [← comp_obj, comp_const, const_obj_obj]
+
+/-- The morphism `R ⟶ S` in `𝒮` obtained by projecting a morphism
+`φ : (ι R).obj a ⟶ (ι S).obj b`. -/
+def proj_map {R S : 𝒮} {a : Fib p R} {b : Fib p S}
+    (φ : (ι R).obj a ⟶ (ι S).obj b) : R ⟶ S :=
+  eqToHom (proj_eq a).symm ≫ (p.map φ) ≫ eqToHom (proj_eq b)
+
+/-- For any homomorphism φ in a fiber Fib S, its image under ι S lies over 𝟙 S -/
+instance homLift {S : 𝒮} {a b : Fib p S} (φ : a ⟶ b) : IsHomLift p (𝟙 S) ((ι S).map φ) := by
+  apply of_fac p _ _ (proj_eq a) (proj_eq b)
+  rw [← Functor.comp_map, Functor.congr_hom (comp_const S)]
+  simp
+
+/-- A version of fullness of the functor `Fib S ⥤ Fiber p S` that can be used inside the category
+`𝒳`. -/
+noncomputable def mapPreimage {S : 𝒮} {a b : Fib p S} (φ : (ι S).obj a ⟶ (ι S).obj b)
+    [IsHomLift p (𝟙 S) φ] : a ⟶ b :=
+  (inducedFunctor _ S).preimage (homMk p S φ)
+
+@[simp]
+lemma mapPreimage_eq {S : 𝒮} {a b : Fib p S} (φ : (ι S).obj a ⟶ (ι S).obj b)
+    [IsHomLift p (𝟙 S) φ] : (ι S).map (mapPreimage φ) = φ := by
+  simp [mapPreimage, congr_hom (inducedFunctor_comp p S)]
+
+/-- The lift of an isomorphism `Φ : (ι S).obj a ≅ (ι S).obj b` lying over `𝟙 S` to an isomorphism
+in `Fib S`. -/
+noncomputable def LiftIso {S : 𝒮} {a b : Fib p S}
+    (Φ : (ι S).obj a ≅ (ι S).obj b) (hΦ : IsHomLift p (𝟙 S) Φ.hom) : a ≅ b := by
+  let a' : Fiber p S := (inducedFunctor p S).obj a
+  let b' : Fiber p S := (inducedFunctor p S).obj b
+  let Φ' : a' ≅ b' := {
+    hom := ⟨Φ.hom, hΦ⟩
+    inv := ⟨Φ.inv, inferInstance⟩
+    hom_inv_id := by
+      ext
+      simp [fiberInclusion.map_comp]
+      simp [fiberInclusion]
+    inv_hom_id := by
+      ext
+      simp [fiberInclusion.map_comp]
+      simp [fiberInclusion]
+    }
+  exact ((inducedFunctor p S).preimageIso Φ')
+
+/-- An object in `Fib p S` isomorphic in `𝒳` to a given object `a : 𝒳` such that `p(a) = S`. -/
+noncomputable def objPreimage {S : 𝒮} {a : 𝒳} (ha : p.obj a = S) : Fib p S :=
+  Functor.objPreimage (inducedFunctor p S) (Fiber.mk ha)
+
+/-- Applying `ι S` to the preimage of `a : 𝒳` in `Fib p S` yields an object isomorphic to `a`. -/
+noncomputable def objObjPreimageIso {S : 𝒮} {a : 𝒳} (ha : p.obj a = S) :
+    (ι S).obj (objPreimage ha) ≅ a :=
+  fiberInclusion.mapIso (Functor.objObjPreimageIso (inducedFunctor p S) (Fiber.mk ha))
+
+instance objObjPreimageIso.IsHomLift {S : 𝒮} {a : 𝒳} (ha : p.obj a = S) :
+    IsHomLift p (𝟙 S) (objObjPreimageIso ha).hom :=
+  (Functor.objObjPreimageIso (inducedFunctor p S) (Fiber.mk ha)).hom.2
+
+section
+
+variable [IsPreFibered p] {R S : 𝒮} {a : 𝒳} (f : R ⟶ S) (ha : p.obj a = S)
+
+/-- The domain, taken in `Fib p R`, of some cartesian morphism lifting a given
+`f : R ⟶ S` in `𝒮` -/
+noncomputable def pullbackObj : Fib p R :=
+  objPreimage (domain_eq p f (IsPreFibered.pullbackMap ha f))
+
+/-- A cartesian morphism lifting `f : R ⟶ S` with domain in the image of `Fib p R` -/
+noncomputable def pullbackMap : (ι R).obj (pullbackObj f ha) ⟶ a :=
+  (objObjPreimageIso (domain_eq p f (IsPreFibered.pullbackMap ha f))).hom ≫
+    (IsPreFibered.pullbackMap ha f)
+
+instance pullbackMap.isCartesian : IsCartesian p f (pullbackMap f ha) := by
+  conv in f => rw [← id_comp f]
+  simp only [id_comp, pullbackMap]
+  infer_instance
+
+end
+
+section
+
+variable {R S : 𝒮} {a : 𝒳} {b b' : Fib p R} (f : R ⟶ S) (ψ : (ι R).obj b' ⟶ a)
+    [IsCartesian p f ψ] (φ : (ι R).obj b ⟶ a) [IsHomLift p f φ]
+
+/-- Given a fibered category p, b' b in Fib R, and a pullback ψ : b ⟶ a in 𝒳, i.e.
+```
+b'       b --ψ--> a
+|        |        |
+v        v        v
+R ====== R --f--> S
+```
+Then the induced map τ : b' ⟶ b can be lifted to the fiber over R -/
+noncomputable def inducedMap : b ⟶ b' :=
+  mapPreimage (IsCartesian.map p f ψ φ)
+
+lemma inducedMap_comp : (ι R).map (inducedMap f ψ φ) ≫ ψ = φ := by
+  simp only [inducedMap, mapPreimage_eq, IsCartesian.fac]
+
+end
+
+section
+
+variable [IsFibered p] {R S : 𝒮} {a : 𝒳} {b : Fib p R}
+
+/-- Given `a : 𝒳`, `b : Fib p R`, and a diagram
+```
+  b --φ--> a
+  -        -
+  |        |
+  v        v
+  R --f--> S
+```
+It can be factorized as
+```
+  b --τ--> b'--ψ--> a
+  -        -        -
+  |        |        |
+  v        v        v
+  R ====== R --f--> S
+```
+with `ψ` cartesian over `f` and `τ` a map in `Fib p R`. -/
+lemma fiber_factorization (ha : p.obj a = S) {b : Fib p R} (f : R ⟶ S) (φ : (ι R).obj b ⟶ a)
+    [IsHomLift p f φ] : ∃ (b' : Fib p R) (τ : b ⟶ b') (ψ : (ι R).obj b' ⟶ a),
+      IsStronglyCartesian p f ψ ∧ (((ι R).map τ) ≫ ψ = φ) :=
+  let ψ := pullbackMap f ha
+  ⟨pullbackObj f ha, inducedMap f ψ φ, ψ, inferInstance, inducedMap_comp f ψ φ⟩
+
+end
+
+end
+
+end HasFibers

Mathlib/CategoryTheory/FiberedCategory/HomLift.lean

@@ -124,7 +124,7 @@ instance comp {R S T : 𝒮} {a b c : 𝒳} (f : R ⟶ S) (g : S ⟶ T) (φ : a
   apply CommSq.horiz_comp (commSq p f φ) (commSq p g ψ)
 
 /-- If `φ : a ⟶ b` and `ψ : b ⟶ c` lift `𝟙 R`, then so does `φ ≫ ψ` -/
-instance lift_id_comp (R : 𝒮) {a b c : 𝒳} (φ : a ⟶ b) (ψ : b ⟶ c)
+instance comp_of_lift_id (R : 𝒮) {a b c : 𝒳} (φ : a ⟶ b) (ψ : b ⟶ c)
     [p.IsHomLift (𝟙 R) φ] [p.IsHomLift (𝟙 R) ψ] : p.IsHomLift (𝟙 R) (φ ≫ ψ) :=
   comp_id (𝟙 R) ▸ comp p (𝟙 R) (𝟙 R) φ ψ
 
@@ -164,22 +164,37 @@ lemma id_lift_eqToHom_codomain {p : 𝒳 ⥤ 𝒮} {R S : 𝒮} (hRS : R = S) {b
     p.IsHomLift (eqToHom hRS) (𝟙 b) := by
   subst hRS hb; simp
 
-instance comp_eqToHom_lift {R S : 𝒮} {a' a b : 𝒳} (f : R ⟶ S) (φ : a ⟶ b) (h : a' = a)
-    [p.IsHomLift f φ] : p.IsHomLift f (eqToHom h ≫ φ) := by
+
+section
+
+variable {R S : 𝒮} {a b : 𝒳} (f : R ⟶ S) (φ : a ⟶ b) [p.IsHomLift f φ]
+
+instance comp_id_lift : p.IsHomLift f (𝟙 a ≫ φ) := by
+  simp_all
+
+instance id_comp_lift  : p.IsHomLift f (φ ≫ 𝟙 b) := by
+  simp_all
+
+instance lift_id_comp : p.IsHomLift (𝟙 R ≫ f) φ := by
+  simp_all
+
+instance lift_comp_id : p.IsHomLift (f ≫ 𝟙 S) φ := by
+  simp_all
+
+instance comp_eqToHom_lift {a' : 𝒳} (h : a' = a) : p.IsHomLift f (eqToHom h ≫ φ) := by
   subst h; simp_all
 
-instance eqToHom_comp_lift {R S : 𝒮} {a b b' : 𝒳} (f : R ⟶ S) (φ : a ⟶ b) (h : b = b')
-    [p.IsHomLift f φ] : p.IsHomLift f (φ ≫ eqToHom h) := by
+instance eqToHom_comp_lift {b' : 𝒳} (h : b = b') : p.IsHomLift f (φ ≫ eqToHom h) := by
   subst h; simp_all
 
-instance lift_eqToHom_comp {R' R S : 𝒮} {a b : 𝒳} (f : R ⟶ S) (φ : a ⟶ b) (h : R' = R)
-    [p.IsHomLift f φ] : p.IsHomLift (eqToHom h ≫ f) φ := by
+instance lift_eqToHom_comp {R' : 𝒮} (h : R' = R) : p.IsHomLift (eqToHom h ≫ f) φ := by
   subst h; simp_all
 
-instance lift_comp_eqToHom {R S S' : 𝒮} {a b : 𝒳} (f : R ⟶ S) (φ : a ⟶ b) (h : S = S')
-    [p.IsHomLift f φ] : p.IsHomLift (f ≫ eqToHom h) φ := by
+instance lift_comp_eqToHom {S' : 𝒮} (h : S = S') : p.IsHomLift (f ≫ eqToHom h) φ := by
   subst h; simp_all
 
+end
+
 @[simp]
 lemma comp_eqToHom_lift_iff {R S : 𝒮} {a' a b : 𝒳} (f : R ⟶ S) (φ : a ⟶ b) (h : a' = a) :
     p.IsHomLift f (eqToHom h ≫ φ) ↔ p.IsHomLift f φ where