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/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import algebraic_topology.simplicial_object
import category_theory.limits.shapes.wide_pullbacks
import category_theory.arrow
/-!
# The Čech Nerve
This file provides a definition of the Čech nerve associated to an arrow, provided
the base category has the correct wide pullbacks.
Several variants are provided, given `f : arrow C`:
1. `f.cech_nerve` is the Čech nerve, considered as a simplicial object in `C`.
2. `f.augmented_cech_nerve` is the augmented Čech nerve, considered as an
augmented simplicial object in `C`.
3. `simplicial_object.cech_nerve` and `simplicial_object.augmented_cech_nerve` are
functorial versions of 1 resp. 2.
-/
open category_theory
open category_theory.limits
noncomputable theory
universes v u
variables {C : Type u} [category.{v} C]
namespace category_theory.arrow
variables (f : arrow C)
variables [∀ n : ℕ, has_wide_pullback.{0} f.right (λ i : fin (n+1), f.left) (λ i, f.hom)]
/-- The Čech nerve associated to an arrow. -/
@[simps]
def cech_nerve : simplicial_object C :=
{ obj := λ n, wide_pullback.{0} f.right
(λ i : fin (n.unop.len + 1), f.left) (λ i, f.hom),
map := λ m n g, wide_pullback.lift (wide_pullback.base _)
(λ i, wide_pullback.π (λ i, f.hom) $ g.unop.to_order_hom i) $ λ j, by simp,
map_id' := λ x, by { ext ⟨⟩, { simpa }, { simp } },
map_comp' := λ x y z f g, by { ext ⟨⟩, { simpa }, { simp } } }
/-- The morphism between Čech nerves associated to a morphism of arrows. -/
@[simps]
def map_cech_nerve {f g : arrow C}
[∀ n : ℕ, has_wide_pullback f.right (λ i : fin (n+1), f.left) (λ i, f.hom)]
[∀ n : ℕ, has_wide_pullback g.right (λ i : fin (n+1), g.left) (λ i, g.hom)]
(F : f ⟶ g) : f.cech_nerve ⟶ g.cech_nerve :=
{ app := λ n, wide_pullback.lift (wide_pullback.base _ ≫ F.right)
(λ i, wide_pullback.π _ i ≫ F.left) $ λ j, by simp,
naturality' := λ x y f, by { ext ⟨⟩, { simp }, { simp } } }
/-- The augmented Čech nerve associated to an arrow. -/
@[simps]
def augmented_cech_nerve : simplicial_object.augmented C :=
{ left := f.cech_nerve,
right := f.right,
hom :=
{ app := λ i, wide_pullback.base _,
naturality' := λ x y f, by { dsimp, simp } } }
/-- The morphism between augmented Čech nerve associated to a morphism of arrows. -/
@[simps]
def map_augmented_cech_nerve {f g : arrow C}
[∀ n : ℕ, has_wide_pullback f.right (λ i : fin (n+1), f.left) (λ i, f.hom)]
[∀ n : ℕ, has_wide_pullback g.right (λ i : fin (n+1), g.left) (λ i, g.hom)]
(F : f ⟶ g) : f.augmented_cech_nerve ⟶ g.augmented_cech_nerve :=
{ left := map_cech_nerve F,
right := F.right,
w' := by { ext, simp } }
end category_theory.arrow
namespace category_theory
namespace simplicial_object
variables [∀ (n : ℕ) (f : arrow C),
has_wide_pullback f.right (λ i : fin (n+1), f.left) (λ i, f.hom)]
/-- The Čech nerve construction, as a functor from `arrow C`. -/
@[simps]
def cech_nerve : arrow C ⥤ simplicial_object C :=
{ obj := λ f, f.cech_nerve,
map := λ f g F, arrow.map_cech_nerve F,
map_id' := λ i, by { ext, { simp }, { simp } },
map_comp' := λ x y z f g, by { ext, { simp }, { simp } } }
/-- The augmented Čech nerve construction, as a functor from `arrow C`. -/
@[simps]
def augmented_cech_nerve : arrow C ⥤ simplicial_object.augmented C :=
{ obj := λ f, f.augmented_cech_nerve,
map := λ f g F, arrow.map_augmented_cech_nerve F,
map_id' := λ x, by { ext, { simp }, { simp }, { refl } },
map_comp' := λ x y z f g, by { ext, { simp }, { simp }, { refl } } }
/-- A helper function used in defining the Čech adjunction. -/
@[simps]
def equivalence_right_to_left (X : simplicial_object.augmented C) (F : arrow C)
(G : X ⟶ F.augmented_cech_nerve) : augmented.to_arrow.obj X ⟶ F :=
{ left := G.left.app _ ≫ wide_pullback.π (λ i, F.hom) 0,
right := G.right,
w' := begin
have := G.w,
apply_fun (λ e, e.app (opposite.op $ simplex_category.mk 0)) at this,
simpa using this,
end }
/-- A helper function used in defining the Čech adjunction. -/
@[simps]
def equivalence_left_to_right (X : simplicial_object.augmented C) (F : arrow C)
(G : augmented.to_arrow.obj X ⟶ F) : X ⟶ F.augmented_cech_nerve :=
{ left :=
{ app := λ x, limits.wide_pullback.lift (X.hom.app _ ≫ G.right)
(λ i, X.left.map (simplex_category.const x.unop i).op ≫ G.left)
(λ i, by { dsimp, erw [category.assoc, arrow.w,
augmented.to_arrow_obj_hom, nat_trans.naturality_assoc,
functor.const_obj_map, category.id_comp] } ),
naturality' := begin
intros x y f,
ext,
{ dsimp,
simp only [wide_pullback.lift_π, category.assoc],
rw [← category.assoc, ← X.left.map_comp],
refl },
{ dsimp,
simp only [functor.const_obj_map, nat_trans.naturality_assoc,
wide_pullback.lift_base, category.assoc],
erw category.id_comp }
end },
right := G.right,
w' := by { ext, dsimp, simp } }
/-- A helper function used in defining the Čech adjunction. -/
@[simps]
def cech_nerve_equiv (X : simplicial_object.augmented C) (F : arrow C) :
(augmented.to_arrow.obj X ⟶ F) ≃ (X ⟶ F.augmented_cech_nerve) :=
{ to_fun := equivalence_left_to_right _ _,
inv_fun := equivalence_right_to_left _ _,
left_inv := begin
intro A,
dsimp,
ext,
{ dsimp,
erw wide_pullback.lift_π,
nth_rewrite 1 ← category.id_comp A.left,
congr' 1,
convert X.left.map_id _,
rw ← op_id,
congr' 1,
ext ⟨a,ha⟩,
change a < 1 at ha,
change 0 = a,
linarith },
{ refl, }
end,
right_inv := begin
intro A,
ext _ ⟨j⟩,
{ dsimp,
simp only [arrow.cech_nerve_map, wide_pullback.lift_π, nat_trans.naturality_assoc],
erw wide_pullback.lift_π,
refl },
{ erw wide_pullback.lift_base,
have := A.w,
apply_fun (λ e, e.app x) at this,
rw nat_trans.comp_app at this,
erw this,
refl },
{ refl }
end }
/-- The augmented Čech nerve construction is right adjoint to the `to_arrow` functor. -/
abbreviation cech_nerve_adjunction :
(augmented.to_arrow : _ ⥤ arrow C) ⊣ augmented_cech_nerve :=
adjunction.mk_of_hom_equiv
{ hom_equiv := cech_nerve_equiv,
hom_equiv_naturality_left_symm' := λ x y f g h, by { ext, { simp }, { simp } },
hom_equiv_naturality_right' := λ x y f g h, by { ext, { simp }, { simp }, { refl } } }
end simplicial_object
end category_theory
namespace category_theory.arrow
variables (f : arrow C)
variables [∀ n : ℕ, has_wide_pushout f.left (λ i : fin (n+1), f.right) (λ i, f.hom)]
/-- The Čech conerve associated to an arrow. -/
@[simps]
def cech_conerve : cosimplicial_object C :=
{ obj := λ n, wide_pushout f.left
(λ i : fin (n.len + 1), f.right) (λ i, f.hom),
map := λ m n g, wide_pushout.desc (wide_pushout.head _)
(λ i, wide_pushout.ι (λ i, f.hom) $ g.to_order_hom i) $
λ i, by { rw [wide_pushout.arrow_ι (λ i, f.hom)] },
map_id' := λ x, by { ext ⟨⟩, { simpa }, { simp } },
map_comp' := λ x y z f g, by { ext ⟨⟩, { simpa }, { simp } } }
/-- The morphism between Čech conerves associated to a morphism of arrows. -/
@[simps]
def map_cech_conerve {f g : arrow C}
[∀ n : ℕ, has_wide_pushout f.left (λ i : fin (n+1), f.right) (λ i, f.hom)]
[∀ n : ℕ, has_wide_pushout g.left (λ i : fin (n+1), g.right) (λ i, g.hom)]
(F : f ⟶ g) : f.cech_conerve ⟶ g.cech_conerve :=
{ app := λ n, wide_pushout.desc (F.left ≫ wide_pushout.head _)
(λ i, F.right ≫ wide_pushout.ι _ i) $
λ i, by { rw [← arrow.w_assoc F, wide_pushout.arrow_ι (λ i, g.hom)] },
naturality' := λ x y f, by { ext, { simp }, { simp } } }
/-- The augmented Čech conerve associated to an arrow. -/
@[simps]
def augmented_cech_conerve : cosimplicial_object.augmented C :=
{ left := f.left,
right := f.cech_conerve,
hom :=
{ app := λ i, wide_pushout.head _,
naturality' := λ x y f, by { dsimp, simp } } }
/-- The morphism between augmented Čech conerves associated to a morphism of arrows. -/
@[simps]
def map_augmented_cech_conerve {f g : arrow C}
[∀ n : ℕ, has_wide_pushout f.left (λ i : fin (n+1), f.right) (λ i, f.hom)]
[∀ n : ℕ, has_wide_pushout g.left (λ i : fin (n+1), g.right) (λ i, g.hom)]
(F : f ⟶ g) : f.augmented_cech_conerve ⟶ g.augmented_cech_conerve :=
{ left := F.left,
right := map_cech_conerve F,
w' := by { ext, simp } }
end category_theory.arrow
namespace category_theory
namespace cosimplicial_object
variables [∀ (n : ℕ) (f : arrow C),
has_wide_pushout f.left (λ i : fin (n+1), f.right) (λ i, f.hom)]
/-- The Čech conerve construction, as a functor from `arrow C`. -/
@[simps]
def cech_conerve : arrow C ⥤ cosimplicial_object C :=
{ obj := λ f, f.cech_conerve,
map := λ f g F, arrow.map_cech_conerve F,
map_id' := λ i, by { ext, { dsimp, simp }, { dsimp, simp } },
map_comp' := λ f g h F G, by { ext, { simp }, { simp } } }
/-- The augmented Čech conerve construction, as a functor from `arrow C`. -/
@[simps]
def augmented_cech_conerve : arrow C ⥤ cosimplicial_object.augmented C :=
{ obj := λ f, f.augmented_cech_conerve,
map := λ f g F, arrow.map_augmented_cech_conerve F,
map_id' := λ f, by { ext, { refl }, { dsimp, simp }, { dsimp, simp } },
map_comp' := λ f g h F G, by { ext, { refl }, { simp }, { simp } } }
/-- A helper function used in defining the Čech conerve adjunction. -/
@[simps]
def equivalence_left_to_right (F : arrow C) (X : cosimplicial_object.augmented C)
(G : F.augmented_cech_conerve ⟶ X) : F ⟶ augmented.to_arrow.obj X :=
{ left := G.left,
right :=
(wide_pushout.ι (λ i, F.hom) 0 ≫ G.right.app (simplex_category.mk 0) : _),
w' := begin
have := G.w,
apply_fun (λ e, e.app (simplex_category.mk 0)) at this,
simpa only [category_theory.functor.id_map, augmented.to_arrow_obj_hom,
wide_pushout.arrow_ι_assoc (λ i, F.hom)],
end }
/-- A helper function used in defining the Čech conerve adjunction. -/
@[simps]
def equivalence_right_to_left (F : arrow C) (X : cosimplicial_object.augmented C)
(G : F ⟶ augmented.to_arrow.obj X) : F.augmented_cech_conerve ⟶ X :=
{ left := G.left,
right := { app := λ x, limits.wide_pushout.desc (G.left ≫ X.hom.app _)
(λ i, G.right ≫ X.right.map (simplex_category.const x i))
begin
rintros j,
rw ← arrow.w_assoc G,
have t := X.hom.naturality (x.const j),
dsimp at t ⊢,
simp only [category.id_comp] at t,
rw ← t,
end,
naturality' := begin
intros x y f,
ext,
{ dsimp,
simp only [wide_pushout.ι_desc_assoc, wide_pushout.ι_desc],
rw [category.assoc, ←X.right.map_comp],
refl },
{ dsimp,
simp only [functor.const_obj_map, ←nat_trans.naturality,
wide_pushout.head_desc_assoc, wide_pushout.head_desc, category.assoc],
erw category.id_comp }
end },
w' := by { ext, simp } }
/-- A helper function used in defining the Čech conerve adjunction. -/
@[simps]
def cech_conerve_equiv (F : arrow C) (X : cosimplicial_object.augmented C) :
(F.augmented_cech_conerve ⟶ X) ≃ (F ⟶ augmented.to_arrow.obj X) :=
{ to_fun := equivalence_left_to_right _ _,
inv_fun := equivalence_right_to_left _ _,
left_inv := begin
intro A,
dsimp,
ext _, { refl }, ext _ ⟨⟩, -- A bug in the `ext` tactic?
{ dsimp,
simp only [arrow.cech_conerve_map, wide_pushout.ι_desc, category.assoc,
← nat_trans.naturality, wide_pushout.ι_desc_assoc],
refl },
{ erw wide_pushout.head_desc,
have := A.w,
apply_fun (λ e, e.app x) at this,
rw nat_trans.comp_app at this,
erw this,
refl },
end,
right_inv := begin
intro A,
ext,
{ refl, },
{ dsimp,
erw wide_pushout.ι_desc,
nth_rewrite 1 ← category.comp_id A.right,
congr' 1,
convert X.right.map_id _,
ext ⟨a,ha⟩,
change a < 1 at ha,
change 0 = a,
linarith },
end }
/-- The augmented Čech conerve construction is left adjoint to the `to_arrow` functor. -/
abbreviation cech_conerve_adjunction :
augmented_cech_conerve ⊣ (augmented.to_arrow : _ ⥤ arrow C) :=
adjunction.mk_of_hom_equiv
{ hom_equiv := cech_conerve_equiv,
hom_equiv_naturality_left_symm' := λ x y f g h, by { ext, { refl }, { simp }, { simp } },
hom_equiv_naturality_right' := λ x y f g h, by { ext, { simp }, { simp } } }
end cosimplicial_object
end category_theory