84 lines
2.6 KiB
Text
84 lines
2.6 KiB
Text
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/-
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Copyright (c) 2016 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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Authors: Floris van Doorn, Egbert Rijke
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-/
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import hit.two_quotient .seq_colim
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open eq prod sum two_quotient sigma function relation e_closure nat seq_colim
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namespace localization
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section quasi_local_extension
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universe variables u v w
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parameters {A : Type.{u}} {P : A → Type.{v}} {Q : A → Type.{w}} (F : Πa, P a → Q a)
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definition is_local [class] (Y : Type) : Type :=
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Π(a : A), is_equiv (λg, g ∘ F a : (Q a → Y) → (P a → Y))
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section
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parameter (X : Type.{max u v w})
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local abbreviation Y := X + Σa, (P a → X) × Q a
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-- do we want to remove the contractible pairs?
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inductive qleR : Y → Y → Type :=
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| J : Π{a : A} (f : P a → X) (p : P a), qleR (inr ⟨a, (f, F a p)⟩) (inl (f p))
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| k : Π{a : A} (g : Q a → X) (q : Q a), qleR (inl (g q)) (inr ⟨a, (g ∘ F a, q)⟩)
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inductive qleQ : Π⦃y₁ y₂ : Y⦄, e_closure qleR y₁ y₂ → e_closure qleR y₁ y₂ → Type :=
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| K : Π{a : A} (g : Q a → X) (p : P a), qleQ [qleR.k g (F a p)] [qleR.J (g ∘ F a) p]⁻¹ʳ
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definition one_step_localization : Type := two_quotient qleR qleQ
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definition incl : X → one_step_localization := incl0 _ _ ∘ inl
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end
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variables (X : Type.{max u v w}) {Z : Type}
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definition n_step_localization : ℕ → Type :=
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nat.rec X (λn Y, localization.one_step_localization F Y)
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definition incln (n : ℕ) :
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n_step_localization X n → n_step_localization X (succ n) :=
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localization.incl F (n_step_localization X n)
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-- localization if P and Q consist of ω-compact types
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definition localization : Type := seq_colim (incln X)
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definition incll : X → localization X := ι' _ 0
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protected definition rec {P : localization X → Type} [Πz, is_local (P z)]
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(H : Πx, P (incll X x)) (z : localization X) : P z :=
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begin
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exact sorry
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end
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definition extend {Y Z : Type} (f : Y → Z) [is_local Z] (x : one_step_localization Y) : Z :=
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begin
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induction x,
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{ induction a,
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{ exact f a},
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{ induction a with a v, induction v with f q, exact sorry}},
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{ exact sorry},
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{ exact sorry}
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end
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protected definition elim {Y : Type} [is_local Y]
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(H : X → Y) (z : localization X) : Y :=
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begin
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induction z with n x n x,
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{ induction n with n IH,
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{ exact H x},
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induction x,
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{ induction a,
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{ exact IH a},
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{ induction a with a v, induction v with f q, exact sorry}},
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{ exact sorry},
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exact sorry},
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exact sorry
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end
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end quasi_local_extension
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end localization
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