135 lines
4.6 KiB
Text
135 lines
4.6 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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Module: hit.colimit
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Authors: Floris van Doorn
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Definition of general colimits and sequential colimits.
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-/
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/- definition of a general colimit -/
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open eq nat type_quotient sigma
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namespace colimit
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context
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parameters {I J : Type} (A : I → Type) (dom cod : J → I)
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(f : Π(j : J), A (dom j) → A (cod j))
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variables {i : I} (a : A i) (j : J) (b : A (dom j))
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local abbreviation B := Σ(i : I), A i
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inductive colim_rel : B → B → Type :=
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| Rmk : Π{j : J} (a : A (dom j)), colim_rel ⟨cod j, f j a⟩ ⟨dom j, a⟩
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open colim_rel
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local abbreviation R := colim_rel
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-- TODO: define this in root namespace
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definition colimit : Type :=
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type_quotient colim_rel
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definition incl : colimit :=
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class_of R ⟨i, a⟩
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abbreviation ι := @incl
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definition cglue : ι (f j b) = ι b :=
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eq_of_rel (Rmk f b)
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protected definition rec {P : colimit → Type}
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(Pincl : Π⦃i : I⦄ (x : A i), P (ι x))
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(Pglue : Π(j : J) (x : A (dom j)), cglue j x ▹ Pincl (f j x) = Pincl x)
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(y : colimit) : P y :=
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begin
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fapply (type_quotient.rec_on y),
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{ intro a, cases a, apply Pincl},
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{ intros [a, a', H], cases H, apply Pglue}
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end
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protected definition rec_on [reducible] {P : colimit → Type} (y : colimit)
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(Pincl : Π⦃i : I⦄ (x : A i), P (ι x))
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(Pglue : Π(j : J) (x : A (dom j)), cglue j x ▹ Pincl (f j x) = Pincl x) : P y :=
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rec Pincl Pglue y
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protected definition elim {P : Type} (Pincl : Π⦃i : I⦄ (x : A i), P)
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(Pglue : Π(j : J) (x : A (dom j)), Pincl (f j x) = Pincl x) (y : colimit) : P :=
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rec Pincl (λj a, !tr_constant ⬝ Pglue j a) y
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protected definition elim_on [reducible] {P : Type} (y : colimit)
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(Pincl : Π⦃i : I⦄ (x : A i), P)
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(Pglue : Π(j : J) (x : A (dom j)), Pincl (f j x) = Pincl x) : P :=
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elim Pincl Pglue y
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definition rec_cglue [reducible] {P : colimit → Type}
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(Pincl : Π⦃i : I⦄ (x : A i), P (ι x))
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(Pglue : Π(j : J) (x : A (dom j)), cglue j x ▹ Pincl (f j x) = Pincl x)
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{j : J} (x : A (dom j)) : apD (rec Pincl Pglue) (cglue j x) = sorry ⬝ Pglue j x ⬝ sorry :=
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sorry
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definition elim_cglue [reducible] {P : Type}
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(Pincl : Π⦃i : I⦄ (x : A i), P)
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(Pglue : Π(j : J) (x : A (dom j)), Pincl (f j x) = Pincl x)
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{j : J} (x : A (dom j)) : ap (elim Pincl Pglue) (cglue j x) = sorry ⬝ Pglue j x ⬝ sorry :=
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sorry
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end
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end colimit
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/- definition of a sequential colimit -/
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namespace seq_colim
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context
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parameters {A : ℕ → Type} (f : Π⦃n⦄, A n → A (succ n))
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variables {n : ℕ} (a : A n)
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local abbreviation B := Σ(n : ℕ), A n
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inductive seq_rel : B → B → Type :=
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| Rmk : Π{n : ℕ} (a : A n), seq_rel ⟨succ n, f a⟩ ⟨n, a⟩
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open seq_rel
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local abbreviation R := seq_rel
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-- TODO: define this in root namespace
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definition seq_colim : Type :=
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type_quotient seq_rel
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definition inclusion : seq_colim :=
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class_of R ⟨n, a⟩
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abbreviation sι := @inclusion
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definition glue : sι (f a) = sι a :=
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eq_of_rel (Rmk f a)
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protected definition rec [reducible] {P : seq_colim → Type}
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(Pincl : Π⦃n : ℕ⦄ (a : A n), P (sι a))
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(Pglue : Π(n : ℕ) (a : A n), glue a ▹ Pincl (f a) = Pincl a) (aa : seq_colim) : P aa :=
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begin
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fapply (type_quotient.rec_on aa),
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{ intro a, cases a, apply Pincl},
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{ intros [a, a', H], cases H, apply Pglue}
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end
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protected definition rec_on [reducible] {P : seq_colim → Type} (aa : seq_colim)
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(Pincl : Π⦃n : ℕ⦄ (a : A n), P (sι a))
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(Pglue : Π⦃n : ℕ⦄ (a : A n), glue a ▹ Pincl (f a) = Pincl a)
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: P aa :=
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rec Pincl Pglue aa
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protected definition elim {P : Type} (Pincl : Π⦃n : ℕ⦄ (a : A n), P)
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(Pglue : Π⦃n : ℕ⦄ (a : A n), Pincl (f a) = Pincl a) : seq_colim → P :=
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rec Pincl (λn a, !tr_constant ⬝ Pglue a)
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protected definition elim_on [reducible] {P : Type} (aa : seq_colim)
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(Pincl : Π⦃n : ℕ⦄ (a : A n), P)
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(Pglue : Π⦃n : ℕ⦄ (a : A n), Pincl (f a) = Pincl a) : P :=
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elim Pincl Pglue aa
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definition rec_glue {P : seq_colim → Type} (Pincl : Π⦃n : ℕ⦄ (a : A n), P (sι a))
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(Pglue : Π⦃n : ℕ⦄ (a : A n), glue a ▹ Pincl (f a) = Pincl a) {n : ℕ} (a : A n)
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: apD (rec Pincl Pglue) (glue a) = sorry ⬝ Pglue a ⬝ sorry :=
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sorry
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definition elim_glue {P : Type} (Pincl : Π⦃n : ℕ⦄ (a : A n), P)
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(Pglue : Π⦃n : ℕ⦄ (a : A n), Pincl (f a) = Pincl a) {n : ℕ} (a : A n)
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: ap (elim Pincl Pglue) (glue a) = sorry ⬝ Pglue a ⬝ sorry :=
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sorry
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end
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end seq_colim
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