274 lines
12 KiB
Text
274 lines
12 KiB
Text
/-
|
|
Copyright (c) 2014 Floris van Doorn. All rights reserved.
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
|
Author: Floris van Doorn
|
|
-/
|
|
|
|
import .basic algebra.relation algebra.binary
|
|
|
|
open eq eq.ops category
|
|
|
|
namespace morphism
|
|
variables {ob : Type} [C : category ob] include C
|
|
variables {a b c : ob} {g : b ⟶ c} {f : a ⟶ b} {h : b ⟶ a}
|
|
inductive is_section [class] (f : a ⟶ b) : Type
|
|
:= mk : ∀{g}, g ∘ f = id → is_section f
|
|
inductive is_retraction [class] (f : a ⟶ b) : Type
|
|
:= mk : ∀{g}, f ∘ g = id → is_retraction f
|
|
inductive is_iso [class] (f : a ⟶ b) : Type
|
|
:= mk : ∀{g}, g ∘ f = id → f ∘ g = id → is_iso f
|
|
|
|
attribute is_iso [multiple_instances]
|
|
|
|
definition retraction_of (f : a ⟶ b) [H : is_section f] : hom b a :=
|
|
is_section.rec (λg h, g) H
|
|
definition section_of (f : a ⟶ b) [H : is_retraction f] : hom b a :=
|
|
is_retraction.rec (λg h, g) H
|
|
definition inverse (f : a ⟶ b) [H : is_iso f] : hom b a :=
|
|
is_iso.rec (λg h1 h2, g) H
|
|
|
|
postfix `⁻¹` := inverse
|
|
|
|
theorem inverse_compose (f : a ⟶ b) [H : is_iso f] : f⁻¹ ∘ f = id :=
|
|
is_iso.rec (λg h1 h2, h1) H
|
|
|
|
theorem compose_inverse (f : a ⟶ b) [H : is_iso f] : f ∘ f⁻¹ = id :=
|
|
is_iso.rec (λg h1 h2, h2) H
|
|
|
|
theorem retraction_compose (f : a ⟶ b) [H : is_section f] : retraction_of f ∘ f = id :=
|
|
is_section.rec (λg h, h) H
|
|
|
|
theorem compose_section (f : a ⟶ b) [H : is_retraction f] : f ∘ section_of f = id :=
|
|
is_retraction.rec (λg h, h) H
|
|
|
|
theorem iso_imp_retraction [instance] (f : a ⟶ b) [H : is_iso f] : is_section f :=
|
|
is_section.mk !inverse_compose
|
|
|
|
theorem iso_imp_section [instance] (f : a ⟶ b) [H : is_iso f] : is_retraction f :=
|
|
is_retraction.mk !compose_inverse
|
|
|
|
theorem id_is_iso [instance] : is_iso (ID a) :=
|
|
is_iso.mk !id_compose !id_compose
|
|
|
|
theorem inverse_is_iso [instance] (f : a ⟶ b) [H : is_iso f] : is_iso (f⁻¹) :=
|
|
is_iso.mk !compose_inverse !inverse_compose
|
|
|
|
theorem left_inverse_eq_right_inverse {f : a ⟶ b} {g g' : hom b a}
|
|
(Hl : g ∘ f = id) (Hr : f ∘ g' = id) : g = g' :=
|
|
calc
|
|
g = g ∘ id : by rewrite id_right
|
|
... = g ∘ f ∘ g' : by rewrite -Hr
|
|
... = (g ∘ f) ∘ g' : by rewrite assoc
|
|
... = id ∘ g' : by rewrite Hl
|
|
... = g' : by rewrite id_left
|
|
|
|
theorem retraction_eq_intro [H : is_section f] (H2 : f ∘ h = id) : retraction_of f = h :=
|
|
left_inverse_eq_right_inverse !retraction_compose H2
|
|
|
|
theorem section_eq_intro [H : is_retraction f] (H2 : h ∘ f = id) : section_of f = h :=
|
|
symm (left_inverse_eq_right_inverse H2 !compose_section)
|
|
|
|
theorem inverse_eq_intro_right [H : is_iso f] (H2 : f ∘ h = id) : f⁻¹ = h :=
|
|
left_inverse_eq_right_inverse !inverse_compose H2
|
|
|
|
theorem inverse_eq_intro_left [H : is_iso f] (H2 : h ∘ f = id) : f⁻¹ = h :=
|
|
symm (left_inverse_eq_right_inverse H2 !compose_inverse)
|
|
|
|
theorem section_eq_retraction (f : a ⟶ b) [Hl : is_section f] [Hr : is_retraction f] :
|
|
retraction_of f = section_of f :=
|
|
retraction_eq_intro !compose_section
|
|
|
|
theorem section_retraction_imp_iso (f : a ⟶ b) [Hl : is_section f] [Hr : is_retraction f]
|
|
: is_iso f :=
|
|
is_iso.mk (subst (section_eq_retraction f) (retraction_compose f)) (compose_section f)
|
|
|
|
theorem inverse_unique (H H' : is_iso f) : @inverse _ _ _ _ f H = @inverse _ _ _ _ f H' :=
|
|
inverse_eq_intro_left !inverse_compose
|
|
|
|
theorem inverse_involutive (f : a ⟶ b) [H : is_iso f] : (f⁻¹)⁻¹ = f :=
|
|
inverse_eq_intro_right !inverse_compose
|
|
|
|
theorem retraction_of_id : retraction_of (ID a) = id :=
|
|
retraction_eq_intro !id_compose
|
|
|
|
theorem section_of_id : section_of (ID a) = id :=
|
|
section_eq_intro !id_compose
|
|
|
|
theorem iso_of_id : (ID a)⁻¹ = id :=
|
|
inverse_eq_intro_left !id_compose
|
|
|
|
theorem composition_is_section [instance] [Hf : is_section f] [Hg : is_section g]
|
|
: is_section (g ∘ f) :=
|
|
is_section.mk
|
|
(calc
|
|
(retraction_of f ∘ retraction_of g) ∘ g ∘ f
|
|
= retraction_of f ∘ retraction_of g ∘ g ∘ f : by rewrite -assoc
|
|
... = retraction_of f ∘ (retraction_of g ∘ g) ∘ f : by rewrite (assoc _ g f)
|
|
... = retraction_of f ∘ id ∘ f : by rewrite retraction_compose
|
|
... = retraction_of f ∘ f : by rewrite id_left
|
|
... = id : by rewrite retraction_compose)
|
|
|
|
theorem composition_is_retraction [instance] [Hf : is_retraction f] [Hg : is_retraction g]
|
|
: is_retraction (g ∘ f) :=
|
|
is_retraction.mk
|
|
(calc
|
|
(g ∘ f) ∘ section_of f ∘ section_of g
|
|
= g ∘ f ∘ section_of f ∘ section_of g : by rewrite -assoc
|
|
... = g ∘ (f ∘ section_of f) ∘ section_of g : by rewrite -assoc
|
|
... = g ∘ id ∘ section_of g : by rewrite compose_section
|
|
... = g ∘ section_of g : by rewrite id_left
|
|
... = id : by rewrite compose_section)
|
|
|
|
theorem composition_is_inverse [instance] [Hf : is_iso f] [Hg : is_iso g] : is_iso (g ∘ f) :=
|
|
!section_retraction_imp_iso
|
|
|
|
structure isomorphic (a b : ob) :=
|
|
(iso : a ⟶ b)
|
|
[is_iso : is_iso iso]
|
|
|
|
infix `≅`:50 := morphism.isomorphic
|
|
|
|
namespace isomorphic
|
|
open relation
|
|
attribute is_iso [instance]
|
|
|
|
theorem refl (a : ob) : a ≅ a := mk id
|
|
theorem symm ⦃a b : ob⦄ (H : a ≅ b) : b ≅ a := mk (inverse (iso H))
|
|
theorem trans ⦃a b c : ob⦄ (H1 : a ≅ b) (H2 : b ≅ c) : a ≅ c := mk (iso H2 ∘ iso H1)
|
|
|
|
theorem is_equivalence_eq [instance] (T : Type) : is_equivalence (isomorphic : ob → ob → Type) :=
|
|
is_equivalence.mk refl symm trans
|
|
end isomorphic
|
|
|
|
inductive is_mono [class] (f : a ⟶ b) : Prop :=
|
|
mk : (∀c (g h : hom c a), f ∘ g = f ∘ h → g = h) → is_mono f
|
|
inductive is_epi [class] (f : a ⟶ b) : Prop :=
|
|
mk : (∀c (g h : hom b c), g ∘ f = h ∘ f → g = h) → is_epi f
|
|
|
|
theorem mono_elim [H : is_mono f] {g h : c ⟶ a} (H2 : f ∘ g = f ∘ h) : g = h :=
|
|
match H with
|
|
is_mono.mk H3 := H3 c g h H2
|
|
end
|
|
|
|
theorem epi_elim [H : is_epi f] {g h : b ⟶ c} (H2 : g ∘ f = h ∘ f) : g = h :=
|
|
match H with
|
|
is_epi.mk H3 := H3 c g h H2
|
|
end
|
|
|
|
theorem section_is_mono [instance] (f : a ⟶ b) [H : is_section f] : is_mono f :=
|
|
is_mono.mk
|
|
(λ c g h H, calc
|
|
g = id ∘ g : by rewrite id_left
|
|
... = (retraction_of f ∘ f) ∘ g : by rewrite -(retraction_compose f)
|
|
... = (retraction_of f ∘ f) ∘ h : by rewrite [-assoc, H, -assoc]
|
|
... = id ∘ h : by rewrite retraction_compose
|
|
... = h : by rewrite id_left)
|
|
|
|
theorem retraction_is_epi [instance] (f : a ⟶ b) [H : is_retraction f] : is_epi f :=
|
|
is_epi.mk
|
|
(λ c g h H, calc
|
|
g = g ∘ id : by rewrite id_right
|
|
... = g ∘ f ∘ section_of f : by rewrite -(compose_section f)
|
|
... = h ∘ f ∘ section_of f : by rewrite [assoc, H, -assoc]
|
|
... = h ∘ id : by rewrite compose_section
|
|
... = h : by rewrite id_right)
|
|
|
|
--these theorems are now proven automatically using type classes
|
|
--should they be instances?
|
|
theorem id_is_mono : is_mono (ID a)
|
|
theorem id_is_epi : is_epi (ID a)
|
|
|
|
theorem composition_is_mono [instance] [Hf : is_mono f] [Hg : is_mono g] : is_mono (g ∘ f) :=
|
|
is_mono.mk
|
|
(λ d h₁ h₂ H,
|
|
have H2 : g ∘ (f ∘ h₁) = g ∘ (f ∘ h₂),
|
|
begin
|
|
rewrite *assoc, exact H
|
|
end,
|
|
mono_elim (mono_elim H2))
|
|
|
|
theorem composition_is_epi [instance] [Hf : is_epi f] [Hg : is_epi g] : is_epi (g ∘ f) :=
|
|
is_epi.mk
|
|
(λ d h₁ h₂ H,
|
|
have H2 : (h₁ ∘ g) ∘ f = (h₂ ∘ g) ∘ f,
|
|
begin
|
|
rewrite -*assoc, exact H
|
|
end,
|
|
epi_elim (epi_elim H2))
|
|
end morphism
|
|
namespace morphism
|
|
--rewrite lemmas for inverses, modified from
|
|
--https://github.com/JasonGross/HoTT-categories/blob/master/theories/Categories/Category/Morphisms.v
|
|
namespace iso
|
|
section
|
|
variables {ob : Type} [C : category ob] include C
|
|
variables {a b c d : ob}
|
|
variables (f : b ⟶ a) (r : c ⟶ d) (q : b ⟶ c) (p : a ⟶ b)
|
|
variables (g : d ⟶ c)
|
|
|
|
variable [Hq : is_iso q] include Hq
|
|
theorem compose_pV : q ∘ q⁻¹ = id := !compose_inverse
|
|
theorem compose_Vp : q⁻¹ ∘ q = id := !inverse_compose
|
|
theorem compose_V_pp : q⁻¹ ∘ (q ∘ p) = p :=
|
|
calc
|
|
q⁻¹ ∘ (q ∘ p) = (q⁻¹ ∘ q) ∘ p : by rewrite assoc
|
|
... = id ∘ p : by rewrite inverse_compose
|
|
... = p : by rewrite id_left
|
|
|
|
theorem compose_p_Vp : q ∘ (q⁻¹ ∘ g) = g :=
|
|
calc
|
|
q ∘ (q⁻¹ ∘ g) = (q ∘ q⁻¹) ∘ g : by rewrite assoc
|
|
... = id ∘ g : by rewrite compose_inverse
|
|
... = g : by rewrite id_left
|
|
|
|
theorem compose_pp_V : (r ∘ q) ∘ q⁻¹ = r :=
|
|
calc
|
|
(r ∘ q) ∘ q⁻¹ = r ∘ q ∘ q⁻¹ : by rewrite assoc
|
|
... = r ∘ id : by rewrite compose_inverse
|
|
... = r : by rewrite id_right
|
|
|
|
theorem compose_pV_p : (f ∘ q⁻¹) ∘ q = f :=
|
|
calc
|
|
(f ∘ q⁻¹) ∘ q = f ∘ q⁻¹ ∘ q : by rewrite assoc
|
|
... = f ∘ id : by rewrite inverse_compose
|
|
... = f : by rewrite id_right
|
|
|
|
theorem inv_pp [H' : is_iso p] : (q ∘ p)⁻¹ = p⁻¹ ∘ q⁻¹ :=
|
|
inverse_eq_intro_left
|
|
(show (p⁻¹ ∘ (q⁻¹)) ∘ q ∘ p = id, from
|
|
by rewrite [-assoc, compose_V_pp, inverse_compose])
|
|
|
|
theorem inv_Vp [H' : is_iso g] : (q⁻¹ ∘ g)⁻¹ = g⁻¹ ∘ q := inverse_involutive q ▸ inv_pp (q⁻¹) g
|
|
theorem inv_pV [H' : is_iso f] : (q ∘ f⁻¹)⁻¹ = f ∘ q⁻¹ := inverse_involutive f ▸ inv_pp q (f⁻¹)
|
|
theorem inv_VV [H' : is_iso r] : (q⁻¹ ∘ r⁻¹)⁻¹ = r ∘ q := inverse_involutive r ▸ inv_Vp q (r⁻¹)
|
|
end
|
|
section
|
|
variables {ob : Type} {C : category ob} include C
|
|
variables {d c b a : ob}
|
|
variables {i : b ⟶ c} {f : b ⟶ a}
|
|
{r : c ⟶ d} {q : b ⟶ c} {p : a ⟶ b}
|
|
{g : d ⟶ c} {h : c ⟶ b}
|
|
{x : b ⟶ d} {z : a ⟶ c}
|
|
{y : d ⟶ b} {w : c ⟶ a}
|
|
variable [Hq : is_iso q] include Hq
|
|
|
|
theorem moveR_Mp (H : y = q⁻¹ ∘ g) : q ∘ y = g := H⁻¹ ▸ compose_p_Vp q g
|
|
theorem moveR_pM (H : w = f ∘ q⁻¹) : w ∘ q = f := H⁻¹ ▸ compose_pV_p f q
|
|
theorem moveR_Vp (H : z = q ∘ p) : q⁻¹ ∘ z = p := H⁻¹ ▸ compose_V_pp q p
|
|
theorem moveR_pV (H : x = r ∘ q) : x ∘ q⁻¹ = r := H⁻¹ ▸ compose_pp_V r q
|
|
theorem moveL_Mp (H : q⁻¹ ∘ g = y) : g = q ∘ y := (moveR_Mp (H⁻¹))⁻¹
|
|
theorem moveL_pM (H : f ∘ q⁻¹ = w) : f = w ∘ q := (moveR_pM (H⁻¹))⁻¹
|
|
theorem moveL_Vp (H : q ∘ p = z) : p = q⁻¹ ∘ z := (moveR_Vp (H⁻¹))⁻¹
|
|
theorem moveL_pV (H : r ∘ q = x) : r = x ∘ q⁻¹ := (moveR_pV (H⁻¹))⁻¹
|
|
theorem moveL_1V (H : h ∘ q = id) : h = q⁻¹ := (inverse_eq_intro_left H)⁻¹
|
|
theorem moveL_V1 (H : q ∘ h = id) : h = q⁻¹ := (inverse_eq_intro_right H)⁻¹
|
|
theorem moveL_1M (H : i ∘ q⁻¹ = id) : i = q := moveL_1V H ⬝ inverse_involutive q
|
|
theorem moveL_M1 (H : q⁻¹ ∘ i = id) : i = q := moveL_V1 H ⬝ inverse_involutive q
|
|
theorem moveR_1M (H : id = i ∘ q⁻¹) : q = i := (moveL_1M (H⁻¹))⁻¹
|
|
theorem moveR_M1 (H : id = q⁻¹ ∘ i) : q = i := (moveL_M1 (H⁻¹))⁻¹
|
|
theorem moveR_1V (H : id = h ∘ q) : q⁻¹ = h := (moveL_1V (H⁻¹))⁻¹
|
|
theorem moveR_V1 (H : id = q ∘ h) : q⁻¹ = h := (moveL_V1 (H⁻¹))⁻¹
|
|
end
|
|
end iso
|
|
|
|
end morphism
|