139 lines
6 KiB
Text
139 lines
6 KiB
Text
/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Floris van Doorn
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Theorems about pullbacks
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-/
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import cubical.square
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open eq equiv is_equiv function equiv.ops prod unit is_trunc sigma
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variables {A₀₀ A₂₀ A₄₀ A₀₂ A₂₂ A₄₂ : Type}
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(f₁₀ : A₀₀ → A₂₀) (f₃₀ : A₂₀ → A₄₀)
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(f₀₁ : A₀₀ → A₀₂) (f₂₁ : A₂₀ → A₂₂) (f₄₁ : A₄₀ → A₄₂)
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(f₁₂ : A₀₂ → A₂₂) (f₃₂ : A₂₂ → A₄₂)
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structure pullback (f₂₁ : A₂₀ → A₂₂) (f₁₂ : A₀₂ → A₂₂) :=
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(pr1 : A₂₀)
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(pr2 : A₀₂)
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(pr1_pr2 : f₂₁ pr1 = f₁₂ pr2)
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namespace pullback
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protected definition sigma_char [constructor] :
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pullback f₂₁ f₁₂ ≃ Σ(a₂₀ : A₂₀) (a₀₂ : A₀₂), f₂₁ a₂₀ = f₁₂ a₀₂ :=
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begin
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fapply equiv.MK,
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{ intro x, induction x with a₂₀ a₀₂ p, exact ⟨a₂₀, a₀₂, p⟩},
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{ intro x, induction x with a₂₀ y, induction y with a₀₂ p, exact pullback.mk a₂₀ a₀₂ p},
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{ intro x, induction x with a₂₀ y, induction y with a₀₂ p, reflexivity},
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{ intro x, induction x with a₂₀ a₀₂ p, reflexivity},
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end
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variables {f₁₀ f₃₀ f₀₁ f₂₁ f₄₁ f₁₂ f₃₂}
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definition pullback_corec [constructor] (p : Πa, f₂₁ (f₁₀ a) = f₁₂ (f₀₁ a)) (a : A₀₀)
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: pullback f₂₁ f₁₂ :=
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pullback.mk (f₁₀ a) (f₀₁ a) (p a)
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definition pullback_eq {x y : pullback f₂₁ f₁₂} (p1 : pr1 x = pr1 y) (p2 : pr2 x = pr2 y)
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(r : square (pr1_pr2 x) (pr1_pr2 y) (ap f₂₁ p1) (ap f₁₂ p2)) : x = y :=
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by induction y; induction x; esimp at *; induction p1; induction p2;
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exact ap (pullback.mk _ _) (eq_of_vdeg_square r)
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definition pullback_comm_equiv [constructor] : pullback f₁₂ f₂₁ ≃ pullback f₂₁ f₁₂ :=
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begin
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fapply equiv.MK,
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{ intro v, induction v with x y p, exact pullback.mk y x p⁻¹},
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{ intro v, induction v with x y p, exact pullback.mk y x p⁻¹},
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{ intro v, induction v, esimp, exact ap _ !inv_inv},
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{ intro v, induction v, esimp, exact ap _ !inv_inv},
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end
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definition pullback_unit_equiv [constructor]
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: pullback (λ(x : A₀₂), star) (λ(x : A₂₀), star) ≃ A₀₂ × A₂₀ :=
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begin
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fapply equiv.MK,
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{ intro v, induction v with x y p, exact (x, y)},
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{ intro v, induction v with x y, exact pullback.mk x y idp},
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{ intro v, induction v, reflexivity},
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{ intro v, induction v, esimp, apply ap _ !is_hprop.elim},
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end
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definition pullback_along {f : A₂₀ → A₂₂} (g : A₀₂ → A₂₂) : pullback f g → A₂₀ :=
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pr1
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postfix `^*`:(max+1) := pullback_along
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variables (f₁₀ f₃₀ f₀₁ f₂₁ f₄₁ f₁₂ f₃₂)
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structure pullback_square (f₁₀ : A₀₀ → A₂₀) (f₁₂ : A₀₂ → A₂₂) (f₀₁ : A₀₀ → A₀₂) (f₂₁ : A₂₀ → A₂₂)
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: Type :=
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(comm : Πa, f₂₁ (f₁₀ a) = f₁₂ (f₀₁ a))
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(is_pullback : is_equiv (pullback_corec comm : A₀₀ → pullback f₂₁ f₁₂))
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attribute pullback_square.is_pullback [instance]
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definition pbs_comm [unfold 9] := @pullback_square.comm
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definition pullback_square_pullback
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: pullback_square (pr1 : pullback f₂₁ f₁₂ → A₂₀) f₁₂ pr2 f₂₁ :=
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pullback_square.mk
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pr1_pr2
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(adjointify _ (λf, f)
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(λf, by induction f; reflexivity)
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(λg, by induction g; reflexivity))
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variables {f₁₀ f₃₀ f₀₁ f₂₁ f₄₁ f₁₂ f₃₂}
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definition pullback_square_equiv [constructor] (s : pullback_square f₁₀ f₁₂ f₀₁ f₂₁)
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: A₀₀ ≃ pullback f₂₁ f₁₂ :=
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equiv.mk _ (pullback_square.is_pullback s)
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definition of_pullback [unfold 9] (s : pullback_square f₁₀ f₁₂ f₀₁ f₂₁)
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(x : pullback f₂₁ f₁₂) : A₀₀ :=
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(pullback_square_equiv s)⁻¹ x
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definition right_of_pullback (s : pullback_square f₁₀ f₁₂ f₀₁ f₂₁)
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(x : pullback f₂₁ f₁₂) : f₁₀ (of_pullback s x) = pr1 x :=
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ap pr1 (to_right_inv (pullback_square_equiv s) x)
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definition down_of_pullback (s : pullback_square f₁₀ f₁₂ f₀₁ f₂₁)
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(x : pullback f₂₁ f₁₂) : f₀₁ (of_pullback s x) = pr2 x :=
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ap pr2 (to_right_inv (pullback_square_equiv s) x)
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-- definition pullback_square_compose_inverse (s : pullback_square f₁₀ f₁₂ f₀₁ f₂₁)
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-- (t : pullback_square f₃₀ f₃₂ f₂₁ f₄₁) (x : pullback f₄₁ (f₃₂ ∘ f₁₂)) : A₀₀ :=
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-- let a₂₀' : pullback f₄₁ f₃₂ :=
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-- pullback.mk (pr1 x) (f₁₂ (pr2 x)) (pr1_pr2 x) in
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-- let a₂₀ : A₂₀ :=
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-- of_pullback t a₂₀' in
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-- have a₀₀' : pullback f₂₁ f₁₂,
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-- from pullback.mk a₂₀ (pr2 x) !down_of_pullback,
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-- show A₀₀,
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-- from of_pullback s a₀₀'
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-- local attribute pullback_square_compose_inverse [reducible]
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-- definition down_psci (s : pullback_square f₁₀ f₁₂ f₀₁ f₂₁)
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-- (t : pullback_square f₃₀ f₃₂ f₂₁ f₄₁) (x : pullback f₄₁ (f₃₂ ∘ f₁₂)) :
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-- f₀₁ (pullback_square_compose_inverse s t x) = pr2 x :=
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-- by apply down_of_pullback
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-- definition pullback_square_compose [constructor] (s : pullback_square f₁₀ f₁₂ f₀₁ f₂₁)
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-- (t : pullback_square f₃₀ f₃₂ f₂₁ f₄₁) : pullback_square (f₃₀ ∘ f₁₀) (f₃₂ ∘ f₁₂) f₀₁ f₄₁ :=
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-- pullback_square.mk
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-- (λa, pbs_comm t (f₁₀ a) ⬝ ap f₃₂ (pbs_comm s a))
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-- (adjointify _
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-- (pullback_square_compose_inverse s t)
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-- begin
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-- intro x, induction x with x y p, esimp,
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-- fapply pullback_eq: esimp,
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-- { exact ap f₃₀ !right_of_pullback ⬝ !right_of_pullback},
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-- { apply down_of_pullback},
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-- { esimp, exact sorry }
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-- end
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-- begin
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-- intro x, esimp, exact sorry
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-- end)
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end pullback
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