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basic.lean
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basic.lean
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import category_theory.limits.shapes.wide_pullbacks
--import category_theory.over
import algebraic_topology.simplicial_object
import category_theory.products.basic
import category_theory.arrow
import for_mathlib.simplicial.constant
noncomputable theory
namespace category_theory
open category_theory.limits
universes v u
variables {C : Type u} [category.{v} C]
namespace limits
-- TODO: Move this! (or just make a mathlib PR!)
abbreviation has_wide_pullback {J : Type v} (B : C) (objs : J → C)
(arrows : Π (j : J), objs j ⟶ B) := has_limit (wide_pullback_shape.wide_cospan B objs arrows)
abbreviation wide_pullback {J : Type v} (B : C) (objs : J → C) (arrows : Π (j : J), objs j ⟶ B)
[has_wide_pullback B objs arrows] : C :=
limit (wide_pullback_shape.wide_cospan B objs arrows)
abbreviation wide_pullback.π {J : Type v} {B : C} {objs : J → C} {arrows : Π (j : J), objs j ⟶ B}
[has_wide_pullback B objs arrows] (j : J) : wide_pullback B objs arrows ⟶ objs j :=
limit.π (wide_pullback_shape.wide_cospan B objs arrows) (option.some j)
abbreviation wide_pullback.base {J : Type v} {B : C} {objs : J → C} {arrows : Π (j : J), objs j ⟶ B}
[has_wide_pullback B objs arrows] : wide_pullback B objs arrows ⟶ B :=
limit.π (wide_pullback_shape.wide_cospan B objs arrows) option.none
@[simp]
lemma wide_pullback.π_base {J : Type v} {B : C} {objs : J → C} {arrows : Π (j : J), objs j ⟶ B}
[has_wide_pullback B objs arrows] (j : J) :
(wide_pullback.π j : wide_pullback B objs arrows ⟶ _) ≫ arrows j = wide_pullback.base :=
by apply (limit.cone (wide_pullback_shape.wide_cospan B objs arrows)).w (wide_pullback_shape.hom.term j)
abbreviation wide_pullback.lift {J : Type v} {B : C} {objs : J → C} {arrows : Π (j : J), objs j ⟶ B}
[has_wide_pullback B objs arrows] {X : C} (f : X ⟶ B) (π : Π (j : J), X ⟶ objs j)
(w : ∀ j, π j ≫ arrows j = f) : X ⟶ wide_pullback B objs arrows :=
limit.lift (wide_pullback_shape.wide_cospan B objs arrows)
(wide_pullback_shape.mk_cone f π (by convert w))
@[simp]
lemma wide_pullback.lift_π {J : Type v} {B : C} {objs : J → C} {arrows : Π (j : J), objs j ⟶ B}
[has_wide_pullback B objs arrows] {X : C} (f : X ⟶ B) (π : Π (j : J), X ⟶ objs j)
(w : ∀ j, π j ≫ arrows j = f) (j : J) : wide_pullback.lift f π w ≫ wide_pullback.π j = π j :=
(limit.is_limit (wide_pullback_shape.wide_cospan B objs arrows)).fac _ _
@[simp]
lemma wide_pullback.lift_base {J : Type v} {B : C} {objs : J → C} {arrows : Π (j : J), objs j ⟶ B}
[has_wide_pullback B objs arrows] {X : C} (f : X ⟶ B) (π : Π (j : J), X ⟶ objs j)
(w : ∀ j, π j ≫ arrows j = f) : wide_pullback.lift f π w ≫ wide_pullback.base = f :=
(limit.is_limit (wide_pullback_shape.wide_cospan B objs arrows)).fac _ _
--@[ext]
lemma wide_pullback.hom_eq_lift {J : Type v} {B : C} {objs : J → C} {arrows : Π (j : J), objs j ⟶ B}
[has_wide_pullback B objs arrows] {X : C} (f : X ⟶ B) (π : Π (j : J), X ⟶ objs j)
(w : ∀ j, π j ≫ arrows j = f) (g : X ⟶ wide_pullback B objs arrows) :
(∀ j : J, g ≫ wide_pullback.π j = π j) → g ≫ wide_pullback.base = f → wide_pullback.lift f π w = g :=
begin
intros h1 h2,
symmetry,
apply (limit.is_limit (wide_pullback_shape.wide_cospan B objs arrows)).uniq
(wide_pullback_shape.mk_cone f π (by convert w)) g,
rintro (j|j),
exact h2,
exact h1 _,
end
lemma wide_pullback.eq_lift {J : Type v} {B : C} {objs : J → C} {arrows : Π (j : J), objs j ⟶ B}
[has_wide_pullback B objs arrows] {X : C} (f : X ⟶ wide_pullback B objs arrows) :
f = wide_pullback.lift (f ≫ wide_pullback.base) (λ j, f ≫ wide_pullback.π _) (by tidy) :=
by {symmetry, apply wide_pullback.hom_eq_lift, tidy}
@[ext]
lemma wide_pullback.hom_ext {J : Type v} {B : C} {objs : J → C} {arrows : Π (j : J), objs j ⟶ B}
[has_wide_pullback B objs arrows] {X : C} (f g : X ⟶ wide_pullback B objs arrows) :
(∀ j : J, f ≫ wide_pullback.π j = g ≫ wide_pullback.π j) →
f ≫ wide_pullback.base = g ≫ wide_pullback.base → f = g :=
begin
intros h1 h2,
rw wide_pullback.eq_lift f,
apply wide_pullback.hom_eq_lift,
tidy,
end
end limits
abbreviation ufin (n : ℕ) := ulift (fin n)
abbreviation ufin.up {n} : fin n → ufin n := _root_.ulift.up
abbreviation ufin.map {m n} (f : fin m → fin n) : ufin m → ufin n :=
λ i, ufin.up $ f i.down
abbreviation ufin.succ {n} : ufin n → ufin (n+1) := ufin.map fin.succ
instance {n} : has_zero (ufin (n+1)) := ⟨ufin.up 0⟩
abbreviation ufin.pred {n} (x : ufin (n+1)) (hx : x ≠ 0) : ufin n :=
ufin.up $ fin.pred x.down (λ c, hx $ equiv.ulift.injective c)
lemma ufin.succ_ne_zero {n} (x : ufin n) : x.succ ≠ 0 :=
λ c, fin.succ_ne_zero _ (equiv.ulift.symm.injective c)
@[simp]
lemma ufin.succ_pred {n} (x : ufin n) : x.succ.pred (ufin.succ_ne_zero _) = x :=
equiv.ulift.injective (by simp)
@[simps]
def cech_obj {X B : C} (f : X ⟶ B)
[∀ (n : ℕ), limits.has_wide_pullback B (λ (i : ufin (n+1)), X) (λ i, f)] :
simplicial_object C :=
{ obj := λ x, limits.wide_pullback B (λ (i : ufin (x.unop.len+1)), X) (λ i, f),
map := λ x y g, limits.wide_pullback.lift limits.wide_pullback.base
(λ i, limits.wide_pullback.π $ ufin.map g.unop.to_preorder_hom i) (by simp) }.
-- tidy gets map_id' and map_comp', but it needs a bit of help due to deterministic timeouts :-(
@[simps]
def cech [∀ (n : ℕ) (B X : C) (f : X ⟶ B), limits.has_wide_pullback B (λ i : ufin (n+1), X) (λ i, f)] :
arrow C ⥤ simplicial_object C :=
{ obj := λ F, cech_obj F.hom,
map := λ F G ff,
{ app := λ x, limits.wide_pullback.lift (limits.wide_pullback.base ≫ ff.right)
(λ i, limits.wide_pullback.π i ≫ ff.left) (by {intros i, simp [← category.assoc]}) },
map_id' := begin
intros f,
ext1,
ext1 j,
apply limits.wide_pullback.hom_ext,
tidy,
end,
map_comp' := begin
intros x y z f g,
ext1,
ext1 j,
apply limits.wide_pullback.hom_ext,
{ intros i,
simp [wide_pullback_shape.mk_cone] },
{ simp [wide_pullback_shape.mk_cone] }
end }.
namespace cech
def augmentation [∀ (n : ℕ) (B X : C) (f : X ⟶ B), limits.has_wide_pullback B (λ i : ufin (n+1), X) (λ i, f)]
(F : arrow C) : cech.obj F ⟶ simplicial_object.const.obj F.right := { app := λ x, limits.wide_pullback.base }
def augmentation_obj {B X : C} (f : X ⟶ B)
[∀ (n : ℕ), limits.has_wide_pullback B (λ i : ufin (n+1), X) (λ i, f)] :
cech_obj f ⟶ simplicial_object.const.obj B := { app := λ x, limits.wide_pullback.base }
open_locale simplicial
def augmentation_obj_iso {B X : C} (f : X ⟶ B)
[∀ (n : ℕ), limits.has_wide_pullback B (λ i : ufin (n+1), X) (λ i, f)] :
(cech_obj f) _[0] ≅ X :=
{ hom := limits.wide_pullback.π 0,
inv := limits.wide_pullback.lift f (λ x, 𝟙 _) (by simp),
hom_inv_id' := begin
ext,
{ have : j = 0,
{ dsimp at j,
ext,
rcases j with ⟨⟨j,hj⟩⟩,
change j = 0,
rw zero_add at hj,
linarith },
rw this,
simp, },
{ simp }
end }
end cech
end category_theory