/
over.lean
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/
over.lean
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import category_theory.comma
import category_theory.adjunction.basic
import category_theory.limits.shapes
import category_theory.epi_mono
import category_theory.limits.over
import category_theory.closed.cartesian
import category.binary_products
import category.adjunction
/-!
# Properties of the over category.
We can interpret the forgetful functor `forget : over B ⥤ C` as dependent sum,
(written `Σ_B`)
and when C has binary products, it has a right adjoint `B*` given by
`A ↦ (π₁ : B × A → B)`, denoted `star` in Lean.
Furthermore, if the original category C has pullbacks and terminal object (i.e.
all finite limits), `B*` has a right adjoint iff `B` is exponentiable in `C`.
This right adjoint is written `Π_B` and is interpreted as dependent product.
Given `f : A ⟶ B` in `C/B`, the iterated slice `(C/B)/f` is isomorphic to
`C/A`.
-/
noncomputable theory
namespace category_theory
open category limits
universes v u
variables {C : Type u} [category.{v} C]
-- def has_finite_products_of_has_finite_limits [has_finite_limits.{v} C] : has_finite_products.{v} C :=
-- λ _ _ _, by resetI; apply_instance
def has_finite_coproducts_of_has_finite_colimits [has_finite_colimits.{v} C] : has_finite_coproducts.{v} C :=
λ _ _ _, by resetI; apply_instance
@[simps]
def over_iso {B : C} {f g : over B} (hl : f.left ≅ g.left) (hw : hl.hom ≫ g.hom = f.hom) : f ≅ g :=
{ hom := over.hom_mk hl.hom, inv := over.hom_mk hl.inv (by simp [iso.inv_comp_eq, hw]) }
section adjunction
variable (B : C)
section
variable [has_pullbacks.{v} C]
@[simps]
def real_pullback {A B : C} (f : A ⟶ B) : over B ⥤ over A :=
{ obj := λ g, over.mk (pullback.snd : pullback g.hom f ⟶ A),
map := λ g h k, over.hom_mk (pullback.lift (pullback.fst ≫ k.left) pullback.snd (by simp [pullback.condition])) (by tidy) }
end
section
variable [has_binary_products.{v} C]
@[simps]
def star : C ⥤ over B :=
{ obj := λ A, over.mk (limits.prod.fst : B ⨯ A ⟶ B),
map := λ X Y f, over.hom_mk (limits.prod.map (𝟙 _) f),
map_id' := λ X,
begin
apply over.over_morphism.ext,
dsimp,
simp,
end,
map_comp' := λ X Y Z f g,
begin
apply over.over_morphism.ext,
dsimp,
ext,
{ rw [limits.prod.map_fst, comp_id, assoc, limits.prod.map_fst, comp_id, limits.prod.map_fst,
comp_id] },
{ rw [limits.prod.map_snd, assoc, limits.prod.map_snd, limits.prod.map_snd_assoc] }
end }
local attribute [tidy] tactic.case_bash
def forget_adj_star : over.forget B ⊣ star B :=
adjunction.mk_of_hom_equiv
{ hom_equiv := λ g A,
{ to_fun := λ f, over.hom_mk (prod.lift g.hom f),
inv_fun := λ k, k.left ≫ limits.prod.snd,
left_inv := λ f, prod.lift_snd _ _,
right_inv := λ k,
begin
ext;
dsimp,
rw prod.lift_fst,
rw ← over.w k,
refl,
rw prod.lift_snd,
end },
hom_equiv_naturality_right' := λ X Y Y' f g,
begin
dsimp,
ext1,
dsimp,
rw prod.lift_map,
rw comp_id,
end,
hom_equiv_naturality_left_symm' := λ X' X Y f g, begin dsimp, rw assoc end }
end
def exponentiable_of_star_is_left_adj [has_finite_products C] (h : is_left_adjoint (star B)) :
exponentiable B :=
{ is_adj :=
{ right := star B ⋙ h.right,
adj := adjunction.comp _ _ h.adj (forget_adj_star B) } }
def dependent_sum {A B : C} (f : A ⟶ B) : over A ⥤ over B :=
(over.iterated_slice_equiv (over.mk f)).inverse ⋙ over.forget _
/--
`over.map f` gives nice definitional equalities but `dependent_sum` makes it easy to prove
adjunction properties
-/
def over_map_iso_dependent_sum {A B : C} (f : A ⟶ B) : dependent_sum f ≅ over.map f :=
begin
refine nat_iso.of_components (λ X, over_iso (iso.refl _) (id_comp _)) (λ X Y g, _),
{ ext1,
change g.left ≫ 𝟙 _ = 𝟙 _ ≫ g.left,
rw [comp_id, id_comp] }
end
def over_map_id {A : C} : over.map (𝟙 A) ≅ 𝟭 _ :=
nat_iso.of_components (λ X, over_iso (iso.refl _) (begin dsimp, simp end)) (λ X Y f, begin ext, dsimp, simp end)
def over_map_comp {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) : over.map (f ≫ g) ≅ over.map f ⋙ over.map g :=
nat_iso.of_components (λ X, over_iso (iso.refl _) (begin dsimp, simp end)) (λ X Y f, begin ext, dsimp, simp end)
-- local attribute [instance] has_finite_wide_pullbacks_of_has_finite_limits
def pullback_along {A B : C} (f : A ⟶ B) [has_binary_products (over B)] : over B ⥤ over A :=
star (over.mk f) ⋙ (over.iterated_slice_equiv _).functor
lemma jointly_mono {A B : C} [has_binary_products (over B)] {f g : over B} (t₁ t₂ : A ⟶ (g ⨯ f).left) :
t₁ ≫ (limits.prod.fst : g ⨯ f ⟶ _).left = t₂ ≫ (limits.prod.fst : g ⨯ f ⟶ _).left →
t₁ ≫ (limits.prod.snd : g ⨯ f ⟶ _).left = t₂ ≫ (limits.prod.snd : g ⨯ f ⟶ _).left →
t₁ = t₂ :=
begin
intros h₁ h₂,
let A' : over B := over.mk (t₂ ≫ (g ⨯ f).hom), -- usually easier in practice to use the second one
have : t₁ ≫ (g ⨯ f).hom = t₂ ≫ (g ⨯ f).hom,
rw [← over.w (limits.prod.fst : g ⨯ f ⟶ _), reassoc_of h₁],
let t₁' : A' ⟶ g ⨯ f := over.hom_mk t₁ this,
let t₂' : A' ⟶ g ⨯ f := over.hom_mk t₂,
suffices : t₁' = t₂',
apply congr_arg comma_morphism.left this,
apply prod.hom_ext;
{ ext1, assumption }
end
def iso_pb {A B : C} (f : A ⟶ B) [has_binary_products (over B)] [has_pullbacks C] :
pullback_along f ≅ real_pullback f :=
begin
refine nat_iso.of_components _ _,
{ intro X,
let p : over B := over.mk (pullback.snd ≫ f : pullback X.hom f ⟶ B),
let q : p ⟶ over.mk f ⨯ X := prod.lift (over.hom_mk pullback.snd rfl) (over.hom_mk pullback.fst pullback.condition),
apply over_iso _ _,
refine ⟨pullback.lift _ _ _, q.left, _, _⟩,
{ apply (limits.prod.snd : over.mk f ⨯ X ⟶ _).left },
{ apply (limits.prod.fst : over.mk f ⨯ X ⟶ _).left },
{ rw [over.w limits.prod.snd, ← over.w limits.prod.fst, over.mk_hom] },
{ apply jointly_mono;
simp [← over.comp_left] },
{ apply pullback.hom_ext;
simp [← over.comp_left] },
{ apply pullback.lift_snd } },
{ intros X Y g,
ext1,
dsimp [pullback_along],
apply pullback.hom_ext,
{ simp only [assoc, pullback.lift_fst, ← over.comp_left, limits.prod.map_snd, pullback.lift_fst_assoc] },
{ simp only [assoc, pullback.lift_snd, ← over.comp_left, limits.prod.map_fst, comp_id] } },
end
section
-- local attribute [instance] over.construct_products.over_binary_product_of_pullback
def radj {A B : C} (f : A ⟶ B) [has_pullbacks C] :
over.map f ⊣ real_pullback f :=
adjunction.mk_of_hom_equiv
{ hom_equiv := λ g h,
{ to_fun := λ X, over.hom_mk (pullback.lift X.left g.hom (over.w X)) (pullback.lift_snd _ _ _),
inv_fun := λ Y,
begin
refine over.hom_mk _ _,
refine Y.left ≫ pullback.fst,
dsimp,
rw [← over.w Y, assoc, pullback.condition, assoc], refl,
end,
left_inv := by tidy,
right_inv := λ Y, by { ext, dsimp, simp, dsimp, rw [pullback.lift_snd, ← over.w Y], refl } } }
-- (((over.mk f).iterated_slice_equiv.symm.to_adjunction.comp _ _ (forget_adj_star _)).of_nat_iso_left (over_map_iso_dependent_sum f)).of_nat_iso_right (iso_pb f)
def pullback_id {A : C} [has_pullbacks C] : real_pullback (𝟙 A) ≅ 𝟭 _ :=
adjunction.right_adjoint_uniq (radj _) (adjunction.id.of_nat_iso_left over_map_id.symm)
def pullback_comp {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [has_pullbacks C] :
real_pullback (f ≫ g) ≅ real_pullback g ⋙ real_pullback f :=
adjunction.right_adjoint_uniq (radj _) (((radj _).comp _ _ (radj _)).of_nat_iso_left (over_map_comp _ _).symm)
instance thing {A B : C} (f : A ⟶ B) [has_pullbacks C] : is_right_adjoint (real_pullback f) :=
⟨_, radj f⟩
end
variables [has_finite_products.{v} C] [has_pullbacks.{v} C]
def Pi_obj [exponentiable B] (f : over B) : C := pullback ((exp B).map f.hom) (internalize_hom (𝟙 B))
@[simps]
private def pi_obj.equiv [exponentiable B] (X : C) (Y : over B) :
((star B).obj X ⟶ Y) ≃ (X ⟶ Pi_obj B Y) :=
{ to_fun := λ f, pullback.lift (cartesian_closed.curry f.left) (terminal.from _)
(by { rw [internalize_hom, comp_id, ← curry_natural_left, ← curry_natural_right,
limits.prod.map_fst, comp_id, over.w f], refl }),
inv_fun := λ g,
begin
apply over.hom_mk _ _,
{ apply (cartesian_closed.uncurry (g ≫ pullback.fst)) },
{ rw [← uncurry_natural_right, assoc, pullback.condition, ← assoc, uncurry_natural_left],
dsimp [internalize_hom],
rw [uncurry_curry, limits.prod.map_fst_assoc, comp_id, comp_id] }
end,
left_inv := λ f, by { ext1, simp },
right_inv := λ g,
by { ext1, { simp }, { apply subsingleton.elim } } }
private lemma pi_obj.natural_equiv [exponentiable B] (X' X : C) (Y : over B) (f : X' ⟶ X) (g : (star B).obj X ⟶ Y) :
(pi_obj.equiv B X' Y).to_fun ((star B).map f ≫ g) = f ≫ (pi_obj.equiv B X Y).to_fun g :=
begin
apply pullback.hom_ext,
{ simp [curry_natural_left] },
{ apply subsingleton.elim }
end
def Pi_functor [exponentiable B] : over B ⥤ C :=
adjunction.right_adjoint_of_equiv (pi_obj.equiv B) (pi_obj.natural_equiv B)
def star_adj_pi_of_exponentiable [exponentiable B] : star B ⊣ Pi_functor B :=
adjunction.adjunction_of_equiv_right _ _
instance star_is_left_adj_of_exponentiable [exponentiable B] : is_left_adjoint (star B) :=
⟨Pi_functor B, star_adj_pi_of_exponentiable B⟩
end adjunction
end category_theory