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import category_theory.limits.concrete_category
universes v u
open category_theory
namespace category_theory.limits
local attribute [instance] concrete_category.has_coe_to_fun concrete_category.has_coe_to_sort
variables {C : Type u} [category.{v} C] [concrete_category.{v} C]
section equalizer
lemma concrete.equalizer_ext {X Y : C} (f g : X ⟶ Y) [has_equalizer f g]
[preserves_limit (parallel_pair f g) (forget C)] (x y : equalizer f g)
(h : equalizer.ι f g x = equalizer.ι f g y) : x = y :=
begin
apply concrete.limit_ext,
rintros (a|a),
{ apply h },
{ rw [← limit.w (parallel_pair f g) walking_parallel_pair_hom.right,
comp_apply, comp_apply, h] }
end
def concrete.equalizer_equiv_aux {X Y : C} (f g : X ⟶ Y) :
(parallel_pair f g ⋙ forget C).sections ≃ { x : X // f x = g x } :=
{ to_fun := λ x, ⟨x.1 walking_parallel_pair.zero, begin
have h1 := x.2 walking_parallel_pair_hom.left,
have h2 := x.2 walking_parallel_pair_hom.right,
dsimp at h1 h2,
erw [h1, h2],
end⟩,
inv_fun := λ x,
{ val := λ j,
match j with
| walking_parallel_pair.zero := x.1
| walking_parallel_pair.one := f x.1
end,
property := begin
dsimp [functor.sections],
rintros (a|a) (b|b) (f|f),
{ simp, },
{ refl },
{ exact x.2.symm },
{ simp },
end },
left_inv := begin
rintros ⟨x,hx⟩,
ext (a|a),
{ refl },
{ change _ = x _,
rw ← hx walking_parallel_pair_hom.left,
refl }
end,
right_inv := by { rintros ⟨_,_⟩, ext, refl } }
noncomputable
def concrete.equalizer_equiv {X Y : C} (f g : X ⟶ Y) [has_equalizer f g]
[preserves_limit (parallel_pair f g) (forget C)] :
↥(equalizer f g) ≃ { x // f x = g x } :=
let h1 := limit.is_limit (parallel_pair f g),
h2 := is_limit_of_preserves (forget C) h1,
E := h2.cone_point_unique_up_to_iso (types.limit_cone_is_limit.{_ v} _) in
E.to_equiv.trans $ concrete.equalizer_equiv_aux _ _
@[simp]
lemma concrete.equalizer_equiv_apply {X Y : C} (f g : X ⟶ Y) [has_equalizer f g]
[preserves_limit (parallel_pair f g) (forget C)] (x : equalizer f g):
(concrete.equalizer_equiv f g x : X) = equalizer.ι f g x := rfl
end equalizer
end category_theory.limits
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